Matrices — Class 12 Maths Notes

Topics covered in Matrices

1. 1.Introduction of Matrices

   What is a Matrix?

   Imagine you have a big grid where you can store numbers in rows and columns, just like a big spreadsheet or a large checkerboard. This grid is what mathematicians call a "matrix." A matrix helps us organize numbers or data in a structured way, making it easier to perform calculations and analyze information.

   Real-Life Example:

   Think about your school timetable. It has days of the week across the top and the different class periods down the side. Each slot in the timetable tells you which subject you have at a certain time on a certain day. This timetable can be represented as a matrix, where each entry shows a specific subject for a specific period and day.

   How Does it Work in Math?

   In mathematics, a matrix is typically written as a rectangular array of numbers enclosed by square brackets. For example, a matrix might look like this:

   [123456789]⎣⎡​147​258​369​⎦⎤​

   This is a 3x3 matrix because it has 3 rows and 3 columns.

   Applications in Real Life and Careers:

   Matrices are extremely useful in various fields:

   • In computer graphics, matrices are used to process and manipulate the images you see on your screen.
   • Economists use matrices to model economic activities and predict economic outcomes.
   • In engineering, they are crucial for simulations and structural analyses.
   • Meteorologists use matrices to model weather patterns and make predictions.

   Hands-on Activity:

   To get a hands-on understanding of matrices, try creating your own matrix with something simple like your weekly routine. List the days of the week across the top and parts of the day (morning, afternoon, evening) on the side. Fill in what you usually do during these times.

   Using Matrices in Careers:

   Knowing how to use matrices can open doors to careers in computer science, engineering, economics, data analysis, and even animation and game design. Every time these professionals need to solve complex problems involving multiple variables, matrices are one of the tools they use.

   Understanding matrices not only enhances your problem-solving skills but also gives you a powerful tool for organizing and interpreting information, which is valuable in any career.

   
   

2. 2.Definition of Matrices

   A matrix is a rectangular array of numbers arranged in rows and columns. It's a way to organize data in a grid that allows for efficient operations and manipulations of numerical values. The numbers inside a matrix are called "elements," and each element is defined by its position in terms of its row and column.
   

   Numeric Examples

   Let's look at three different examples of matrices with explanations to help you understand how they can vary in size and content:

   Example 1: A 2x3 Matrix

   [123456][14​25​36​]

   This matrix has 2 rows and 3 columns. It can represent data like coordinates in 2D space for 3 points or other sets of numerical information.

   Example 2: A 3x3 Matrix (Square Matrix)

   [789101112131415]⎣⎡​71013​81114​91215​⎦⎤​

   This is a square matrix because the number of rows and columns are equal. Square matrices are particularly important in mathematics for operations like finding determinants and working with linear transformations.

   Example 3: A 4x1 Matrix (Column Matrix)

   [20304050]⎣⎡​20304050​⎦⎤​

   This matrix is an example of a column matrix because it has only one column. Column matrices are often used to represent vectors in various mathematical and engineering applications.

   These examples show how matrices can be constructed and how their structure can vary depending on the application. Each matrix serves as a useful tool in different areas of mathematics and applied fields, providing a compact way to handle and manipulate multiple data points simultaneously.

3. 3.Order of a Matrix

   What is the Order of a Matrix?

   The order of a matrix refers to the dimensions of the matrix, given by the number of rows and columns it contains. It is usually expressed as "m x n" (read as "m by n"), where "m" represents the number of rows, and "n" represents the number of columns. Understanding the order is crucial for determining the compatibility of matrices for certain operations like addition and multiplication.

   Examples of Matrix Order

   Example 1: A 2x2 Matrix

   [5678][57​68​]

   This matrix has an order of 2x2 because it contains 2 rows and 2 columns. This is often called a square matrix because both dimensions are equal.

   Example 2: A 3x4 Matrix

   [123456789101112]⎣⎡​159​2610​3711​4812​⎦⎤​

   This matrix has an order of 3x4, meaning it has 3 rows and 4 columns. This kind of matrix is often used in applications requiring more data in one dimension than the other, such as data recording different variables over time.

   Example 3: A 1x5 Matrix (Row Matrix)

   [246810][2​4​6​8​10​]

   This is a 1x5 matrix, consisting of one row and five columns. It’s also known as a row matrix. Row matrices can represent a single dataset or vector in a mathematical model or analysis.

   Applying Matrix Order in Operations:

   Knowing the order of matrices is essential for matrix operations. For example:

   • Matrix Addition: You can only add two matrices if they have the same order. The elements are added corresponding row-wise and column-wise.
   • Matrix Multiplication: The number of columns in the first matrix must match the number of rows in the second matrix. If you have a 3x4 matrix and a 4x2 matrix, you can multiply them to get a 3x2 matrix.

   These principles ensure that operations are mathematically possible and meaningful in the context of linear algebra and its applications.

4. 4.Types of Matrices

   Types of Matrices

   In mathematics, matrices can be classified into various types based on their shape, the elements they contain, and their properties. Understanding these types can help you identify the best way to use them in different mathematical operations and real-world applications. Let's explore some of the most common types of matrices:

   1. Square Matrix

   A square matrix has the same number of rows and columns (n x n). It is called "square" because the dimensions form a perfect square. This type of matrix is significant in mathematics for operations like finding the determinant and eigenvalues.

   Example:

   [357169284]⎣⎡​312​568​794​⎦⎤​

   This is a 3x3 square matrix.

   2. Rectangular Matrix

   A rectangular matrix has a different number of rows and columns (m x n), where m ≠ n. This is the most general form of a matrix.

   Example:

   [123456][14​25​36​]

   This is a 2x3 rectangular matrix.

   3. Diagonal Matrix

   A diagonal matrix is a square matrix where all elements outside the main diagonal are zero. The main diagonal itself may contain non-zero elements.

   Example:

   [700080009]⎣⎡​700​080​009​⎦⎤​

   This matrix is diagonal.

   4. Identity Matrix

   An identity matrix is a special kind of diagonal matrix where all the elements on the main diagonal are 1, and all other elements are 0. It is denoted by 𝐼𝑛In​, where n is the size of the matrix.

   Example:

   [100010001]⎣⎡​100​010​001​⎦⎤​

   This is a 3x3 identity matrix.

   5. Zero Matrix

   A zero matrix is a matrix all of whose elements are zero. It can be of any size (m x n).

   Example:

   [000000][00​00​00​]

   This is a 2x3 zero matrix.

   6. Symmetric Matrix

   A symmetric matrix is a square matrix that is equal to its transpose. This means that the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column.

   Example:

   [234356468]⎣⎡​234​356​468​⎦⎤​

   This matrix is symmetric.

   7. Skew-Symmetric Matrix

   A skew-symmetric matrix is a square matrix that is equal to the negative of its transpose. This means that each element at the i-th row and j-th column is the negative of the element at the j-th row and i-th column, and all diagonal elements are zero.

   Example:

   [0−3−630−9690]⎣⎡​036​−309​−6−90​⎦⎤​

   This matrix is skew-symmetric.

   Each type of matrix has unique properties and applications, making them useful in different areas of mathematics, engineering, physics, and computer science. Understanding these types can enhance your problem-solving skills and help you understand more complex mathematical concepts.

5. 5.Exercise Questions

   1. In the matrix 𝐴=[2519−735−251231−517]A=⎣⎡​2353​​5−21​195−5​−71217​⎦⎤​, write: (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements 𝑎13a13​, 𝑎21a21​, 𝑎32a32​, 𝑎43a43​.

      Solution:

      1. (i) The Order of the Matrix: The order of a matrix is defined by the number of rows and columns it has. In this case, the matrix 𝐴A has 3 rows and 4 columns, so the order of the matrix is 3×43×4.

      (ii) The Number of Elements: The number of elements in a matrix is found by multiplying the number of rows by the number of columns. For matrix 𝐴A, with an order of 3×43×4, the number of elements is 3×4=123×4=12.

      (iii) The Elements:

      • 𝑎13a13​ is the element in the 1st row and 3rd column, which is 1919.
      • 𝑎21a21​ is the element in the 2nd row and 1st column, which is 3535.
      • 𝑎32a32​ is the element in the 3rd row and 2nd column, which is 11.
      • 𝑎43a43​ does not exist as there is no 4th row in this matrix.

   2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

   Solution:

   1. Possible Orders for a Matrix with 24 Elements: For a matrix to have 24 elements, the product of the number of rows and columns must equal 24. Possible orders include:

   • 1×241×24 or 24×124×1 (row or column matrix)
   • 2×122×12 or 12×212×2
   • 3×83×8 or 8×38×3
   • 4×64×6 or 6×46×4
   • 24×124×1 (since we already mentioned 1×241×24, this is just the transpose)

   Possible Orders for a Matrix with 13 Elements: Since 13 is a prime number, the only possible orders for a matrix with 13 elements are:

   • 1×131×13 or 13×113×1 (row or column matrix)

   No square or rectangular matrix with more than 1 row and 1 column can have 13 elements because there is no pair of factors other than 1 and 13 that can multiply to give 13.

6. 6.Exercise Questions

   Construct a 2 x 2 matrix, 𝐴=[𝑎𝑖𝑗]A=[aij​], whose elements are given by:

   (i) 𝑎𝑖𝑗=(𝑖+𝑗)22aij​=2(i+j)2​

   (ii) 𝑎𝑖𝑗=𝑖𝑗aij​=ji​

   (iii) 𝑎𝑖𝑗=(𝑖+2𝑗)22aij​=2(i+2j)2​"

   Solution : Let's solve each of these one by one:

   For (i) 𝑎𝑖𝑗=(𝑖+𝑗)22aij​=2(i+j)2​:

   To construct this matrix, we plug in the values of 𝑖i and 𝑗j for a 2x2 matrix, which means 𝑖i and 𝑗j can each be 1 or 2.

   Here are the calculations:

   • When 𝑖=1i=1 and 𝑗=1j=1: 𝑎11=(1+1)22=42=2a11​=2(1+1)2​=24​=2
   • When 𝑖=1i=1 and 𝑗=2j=2: 𝑎12=(1+2)22=92=4.5a12​=2(1+2)2​=29​=4.5
   • When 𝑖=2i=2 and 𝑗=1j=1: 𝑎21=(2+1)22=92=4.5a21​=2(2+1)2​=29​=4.5
   • When 𝑖=2i=2 and 𝑗=2j=2: 𝑎22=(2+2)22=162=8a22​=2(2+2)2​=216​=8

   So the matrix for (i) is:

   𝐴(𝑖)=[24.54.58]A(i)​=[24.5​4.58​]

   For (ii) 𝑎𝑖𝑗=𝑖𝑗aij​=ji​:

   Again, we plug in the values of 𝑖i and 𝑗j:

   • When 𝑖=1i=1 and 𝑗=1j=1: 𝑎11=11=1a11​=11​=1
   • When 𝑖=1i=1 and 𝑗=2j=2: 𝑎12=12=0.5a12​=21​=0.5
   • When 𝑖=2i=2 and 𝑗=1j=1: 𝑎21=21=2a21​=12​=2
   • When 𝑖=2i=2 and 𝑗=2j=2: 𝑎22=22=1a22​=22​=1

   So the matrix for (ii) is:

   𝐴(𝑖𝑖)=[10.521]A(ii)​=[12​0.51​]

   For (iii) 𝑎𝑖𝑗=(𝑖+2𝑗)22aij​=2(i+2j)2​:

   And finally, we plug in the values for this formula:

   • When 𝑖=1i=1 and 𝑗=1j=1: 𝑎11=(1+2⋅1)22=92=4.5a11​=2(1+2⋅1)2​=29​=4.5
   • When 𝑖=1i=1 and 𝑗=2j=2: 𝑎12=(1+2⋅2)22=252=12.5a12​=2(1+2⋅2)2​=225​=12.5
   • When 𝑖=2i=2 and 𝑗=1j=1: 𝑎21=(2+2⋅1)22=162=8a21​=2(2+2⋅1)2​=216​=8
   • When 𝑖=2i=2 and 𝑗=2j=2: 𝑎22=(2+2⋅2)22=362=18a22​=2(2+2⋅2)2​=236​=18

   So the matrix for (iii) is:

   𝐴(𝑖𝑖𝑖)=[4.512.5818]A(iii)​=[4.58​12.518​]

   And that's how we construct these matrices. Each one uses a different formula to determine the values of its elements. This process illustrates how mathematics can be very systematic and also how formulas can be used to generate structured numerical arrangements like matrices!

7. 7.Find the value of a, b, c, and d from the equation:

   Find the value of a, b, c, and d from the equation: [𝑎−𝑏2𝑎+𝑐2𝑎−𝑏3𝑐+𝑑]=[−15013][a−b2a−b​2a+c3c+d​]=[−10​513​]

   Solution: To find the values of a, b, c, and d, we will equate the corresponding elements from both sides of the equation.

   From the first matrix's first row, first column (1,1) and the second matrix's (1,1), we get: 𝑎−𝑏=−1a−b=−1 (1)

   From the first matrix's first row, second column (1,2) and the second matrix's (1,2), we get: 2𝑎+𝑐=52a+c=5 (2)

   From the first matrix's second row, first column (2,1) and the second matrix's (2,1), we get: 2𝑎−𝑏=02a−b=0 (3)

   From the first matrix's second row, second column (2,2) and the second matrix's (2,2), we get: 3𝑐+𝑑=133c+d=13 (4)

   Now we have four equations with four variables. Let's solve these equations step by step.

   Equation (3) gives us: 2𝑎−𝑏=02a−b=0 2𝑎=𝑏2a=b (5)

   Using (5) in (1), we can find the value of a: 𝑎−2𝑎=−1a−2a=−1 −𝑎=−1−a=−1 𝑎=1a=1 (6)

   Now we substitute the value of a in (5) to get b: 2(1)=𝑏2(1)=b 𝑏=2b=2 (7)

   Next, we substitute the value of a in (2) to get c: 2(1)+𝑐=52(1)+c=5 2+𝑐=52+c=5 𝑐=3c=3 (8)

   Finally, we substitute the value of c in (4) to find d: 3(3)+𝑑=133(3)+d=13 9+𝑑=139+d=13 𝑑=4d=4 (9)

   So the values are: a = 1, b = 2, c = 3, and d = 4.

   This problem is a good example of how algebra can be used in matrix equations, which is quite handy in fields like economics, physics, and computer science for solving systems of equations.

8. 8.Exercise Questions

   Question - A matrix 𝐴=[𝑎𝑖𝑗]A=[aij​] is a square matrix if:

   (A) 𝑚<𝑛m<n (B) 𝑚>𝑛m>n (C) 𝑚=𝑛m=n (D) None of these

   Solution: A square matrix is defined as a matrix with the same number of rows and columns. Therefore, for a matrix to be square, the number of rows (m) must be equal to the number of columns (n).

   So, the correct answer is: (C) 𝑚=𝑛m=n

   In practical terms, square matrices are very important in mathematics and are used in various fields such as physics, engineering, and computer science for operations such as rotations, reflections, and determining systems' properties like stability and invertibility.

9. 9.Operations on Matrices

   Matrices are incredibly useful in mathematics, particularly because they can be added, subtracted, and multiplied in ways that have important applications in areas such as linear algebra, physics, computer graphics, and more. Here are the primary operations you can perform on matrices:
   

   1. Matrix Addition and Subtraction

   • Matrix Addition: Two matrices can be added if and only if they have the same dimensions. The addition is done element-wise. This means that each element of one matrix is added to the corresponding element of the second matrix.

     Example: If 𝐴A and 𝐵B are both 2x2 matrices:

     𝐴=[𝑎𝑏𝑐𝑑],𝐵=[𝑒𝑓𝑔ℎ]A=[ac​bd​],B=[eg​fh​]

     Then their sum 𝐶=𝐴+𝐵C=A+B is:

     𝐶=[𝑎+𝑒𝑏+𝑓𝑐+𝑔𝑑+ℎ]C=[a+ec+g​b+fd+h​]

   • Matrix Subtraction: Similar to addition, matrices must be of the same dimensions. Each element of one matrix is subtracted from the corresponding element of the second matrix.

     Example: If 𝐴A and 𝐵B are as defined above, then 𝐶=𝐴−𝐵C=A−B is:

     𝐶=[𝑎−𝑒𝑏−𝑓𝑐−𝑔𝑑−ℎ]C=[a−ec−g​b−fd−h​]

   2. Scalar Multiplication

   • Scalar Multiplication: A matrix can be multiplied by a scalar (a real number). In this operation, every element of the matrix is multiplied by the scalar.

     Example: If 𝐴A is as defined above and 𝑘k is a scalar, then 𝑘𝐴kA is:

     𝑘𝐴=[𝑘𝑎𝑘𝑏𝑘𝑐𝑘𝑑]kA=[kakc​kbkd​]

   3. Matrix Multiplication

   • Matrix Multiplication: Two matrices 𝐴A and 𝐵B can be multiplied if the number of columns in 𝐴A is equal to the number of rows in 𝐵B. The element at the 𝑖i-th row and 𝑗j-th column of the product matrix 𝐴𝐵AB is calculated as the dot product of the 𝑖i-th row of 𝐴A and the 𝑗j-th column of 𝐵B.

     Example: If 𝐴A is an 𝑚×𝑛m×n matrix and 𝐵B is an 𝑛×𝑝n×p matrix, then their product 𝐴𝐵AB is an 𝑚×𝑝m×p matrix. The element 𝑐𝑖𝑗cij​ of 𝐴𝐵AB is computed as:

     𝑐𝑖𝑗=𝑎𝑖1𝑏1𝑗+𝑎𝑖2𝑏2𝑗+…+𝑎𝑖𝑛𝑏𝑛𝑗cij​=ai1​b1j​+ai2​b2j​+…+ain​bnj​

   4. Matrix Transposition

   • Matrix Transposition: Transposing a matrix means flipping a matrix over its diagonal. The rows become columns and vice versa. The transpose of matrix 𝐴A is denoted by 𝐴𝑇AT.

     Example: If 𝐴A is as defined above, then 𝐴𝑇AT is:

     𝐴𝑇=[𝑎𝑐𝑏𝑑]AT=[ab​cd​]

   5. Determinant and Inverse

   • Determinant: The determinant is a special scalar value that can be calculated from a square matrix. It is denoted by det⁡(𝐴)det(A) or ∣𝐴∣∣A∣, and is used to determine if a matrix has an inverse, among other applications.

   • Inverse: The inverse of a matrix 𝐴A is denoted by 𝐴−1A−1 and is a unique matrix such that when it is multiplied by 𝐴A, it yields the identity matrix. Not all matrices have inverses; a matrix must be square and have a non-zero determinant to have an inverse.

   These operations form the basis of many more complex matrix manipulations and have vast applications across quantitative disciplines. Understanding and applying these operations allow us to solve systems of linear equations, transform geometric objects in space, and much more.

10. 10.Find each of the following:

    The matrices given are:

    𝐴=[2432]A=[23​42​], 𝐵=[13−25]B=[1−2​35​], 𝐶=[−2534]C=[−23​54​].

    We are to find:

    (i) 𝐴+𝐵A+B (ii) 𝐴−𝐵A−B (iii) 3𝐴−𝐶3A−C (iv) 𝐴𝐵AB (v) 𝐵𝐴BA

    (i) To find 𝐴+𝐵A+B, we simply add the corresponding elements of matrices A and B:

    𝐴+𝐵=[2+14+33+(−2)2+5]=[3717]A+B=[2+13+(−2)​4+32+5​]=[31​77​].

    Let's continue with the next one.

    Here are the results of the matrix operations:

    (ii) 𝐴−𝐵A−B is calculated by subtracting the corresponding elements of B from A:

    𝐴−𝐵=[2−14−33−(−2)2−5]=[115−3]A−B=[2−13−(−2)​4−32−5​]=[15​1−3​].

    (iii) For 3𝐴−𝐶3A−C, we first multiply each element of A by 3 and then subtract the corresponding elements of C:

    3𝐴−𝐶=3×[2432]−[−2534]=[61296]−[−2534]=[8762]3A−C=3×[23​42​]−[−23​54​]=[69​126​]−[−23​54​]=[86​72​].

    (iv) The product 𝐴𝐵AB is found by taking the dot product of the rows of A with the columns of B:

    𝐴𝐵=[2⋅1+4⋅(−2)2⋅3+4⋅53⋅1+2⋅(−2)3⋅3+2⋅5]=[−626−119]AB=[2⋅1+4⋅(−2)3⋅1+2⋅(−2)​2⋅3+4⋅53⋅3+2⋅5​]=[−6−1​2619​].

    (v) Similarly, 𝐵𝐴BA is found by taking the dot product of the rows of B with the columns of A:

    𝐵𝐴=[1⋅2+3⋅31⋅4+3⋅2(−2)⋅2+5⋅3(−2)⋅4+5⋅2]=[1110112]BA=[1⋅2+3⋅3(−2)⋅2+5⋅3​1⋅4+3⋅2(−2)⋅4+5⋅2​]=[1111​102​].

11. 11.Compute the following:

    The matrix operation, an addition of two 2x2 matrices:

    [𝑎𝑏−𝑏𝑎]+[𝑎𝑏−𝑏𝑎][a−b​ba​]+[a−b​ba​]

    To add these matrices, we add the corresponding entries from each matrix to get a new matrix where each entry is the sum of the corresponding entries from the original matrices.

    Let's compute the addition:

    For the element in the first row and first column, we add 𝑎+𝑎a+a. For the element in the first row and second column, we add 𝑏+𝑏b+b. For the element in the second row and first column, we add −𝑏+(−𝑏)−b+(−b). For the element in the second row and second column, we add 𝑎+𝑎a+a.

    Performing these additions, we get:

    [𝑎+𝑎𝑏+𝑏−𝑏+(−𝑏)𝑎+𝑎]=[2𝑎2𝑏−2𝑏2𝑎][a+a−b+(−b)​b+ba+a​]=[2a−2b​2b2a​]

    So, the result of the matrix addition is:

    [2𝑎2𝑏−2𝑏2𝑎][2a−2b​2b2a​]

    This is the final answer.

12. 12.Compute the following:

    [cos2xsin2xsin2xcos2x]+[sin2xcos2xcos2xsin2x]

    Solution Matrix 1: [cos⁡2𝑥sin⁡2𝑥sin⁡2𝑥cos⁡2𝑥][cos2xsin2x​sin2xcos2x​]

    Matrix 2: [sin⁡2𝑥cos⁡2𝑥cos⁡2𝑥sin⁡2𝑥][sin2xcos2x​cos2xsin2x​]

    To add these two matrices, we add the corresponding elements from each matrix to get a new matrix:

    Resulting Matrix: [(cos⁡2𝑥+sin⁡2𝑥)(sin⁡2𝑥+cos⁡2𝑥)(sin⁡2𝑥+cos⁡2𝑥)(cos⁡2𝑥+sin⁡2𝑥)][(cos2x+sin2x)(sin2x+cos2x)​(sin2x+cos2x)(cos2x+sin2x)​]

    Simplifying using the Pythagorean identity sin⁡2𝑥+cos⁡2𝑥=1sin2x+cos2x=1, the resulting matrix is:

    [1111][11​11​]

    This is the final result of the addition of the two matrices.

13. 13.Compute the following:

    
    
    

    A step-by-step procedure to compute the matrix operations 𝐴+𝐵A+B, 𝐵−𝐶B−C, and to verify the matrix identity 𝐴+(𝐵−𝐶)=(𝐴+𝐵)−𝐶A+(B−C)=(A+B)−C with the given matrices A, B, and C. Let's solve this step by step.

    Step 1: Define the Matrices 𝐴=[12−35021−11]A=⎣⎡​151​20−1​−321​⎦⎤​ 𝐵=[3−12−425203]B=⎣⎡​3−42​−120​253​⎦⎤​ 𝐶=[4120321−23]C=⎣⎡​401​13−2​223​⎦⎤​

    Step 2: Compute 𝐴+𝐵A+B We add corresponding elements from matrices A and B: 𝐴+𝐵=[1+32+(−1)−3+25+(−4)0+22+51+2−1+01+3]A+B=⎣⎡​1+35+(−4)1+2​2+(−1)0+2−1+0​−3+22+51+3​⎦⎤​ 𝐴+𝐵=[41−11273−14]A+B=⎣⎡​413​12−1​−174​⎦⎤​

    Step 3: Compute 𝐵−𝐶B−C We subtract corresponding elements of C from B: 𝐵−𝐶=[3−4−1−12−2−4−02−35−22−10−(−2)3−3]B−C=⎣⎡​3−4−4−02−1​−1−12−30−(−2)​2−25−23−3​⎦⎤​ 𝐵−𝐶=[−1−20−4−13120]B−C=⎣⎡​−1−41​−2−12​030​⎦⎤​

    Step 4: Verify the Identity 𝐴+(𝐵−𝐶)=(𝐴+𝐵)−𝐶A+(B−C)=(A+B)−C Compute 𝐴+(𝐵−𝐶)A+(B−C) using the result from Step 3: 𝐴+(𝐵−𝐶)=𝐴+[−1−20−4−13120]A+(B−C)=A+⎣⎡​−1−41​−2−12​030​⎦⎤​ 𝐴+(𝐵−𝐶)=[1+(−1)2+(−2)−3+05+(−4)0+(−1)2+31+1−1+21+0]
    
    A+(B−C)=⎣⎡​1+(−1)5+(−4)1+1​2+(−2)0+(−1)−1+2​−3+02+31+0​⎦⎤​ 𝐴+(𝐵−𝐶)=[00−31−15211]
    
    A+(B−C)=⎣⎡​012​0−11​−351​⎦⎤​

    Now, compute (𝐴+𝐵)−𝐶(A+B)−C using the result from Step 2: (𝐴+𝐵)−𝐶=[41−11273−14]−𝐶(A+B)−C=⎣⎡​413​12−1​−174​⎦⎤​−C (𝐴+𝐵)−𝐶=[4−41−1−1−21−02−37−23−1−1−(−2)4−3]
    
    (A+B)−C=⎣⎡​4−41−03−1​1−12−3−1−(−2)​−1−27−24−3​⎦⎤​ (𝐴+𝐵)−𝐶=[00−31−15211]
    
    (A+B)−C=⎣⎡​012​0−11​−351​⎦⎤​

    Step 5: Comparing Results for Verification Comparing the matrices from Step 4, we see that 𝐴+(𝐵−𝐶)A+(B−C) and (𝐴+𝐵)−𝐶(A+B)−C are indeed equal, thus verifying the identity.

14. 14.Exercise Questions

    Find 𝑋X, if 𝑌=[3214]Y=[31​24​] and 2𝑋+𝑌=[10−32]2X+Y=[1−3​02​].

    Now, let's solve for the matrix 𝑋X.

    To find 𝑋X, we need to isolate it in the equation 2𝑋+𝑌=[10−32]2X+Y=[1−3​02​]. This can be done by subtracting 𝑌Y from the right-hand side matrix and then dividing the result by 2. The calculation proceeds as follows:

    2𝑋=[10−32]−𝑌2X=[1−3​02​]−Y

    2𝑋=[10−32]−[3214]2X=[1−3​02​]−[31​24​]

    2𝑋=[1−30−2−3−12−4]2X=[1−3−3−1​0−22−4​]

    2𝑋=[−2−2−4−2]2X=[−2−4​−2−2​]

    Now, divide by 2 to find 𝑋X:

    𝑋=12[−2−2−4−2]X=21​[−2−4​−2−2​]

    𝑋=[−1−1−2−1]X=[−1−2​−1−1​]

    So, the matrix 𝑋X is:

    [−1−1−2−1][−1−2​−1−1​]

15. 15.A trust fund has ₹30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide ₹30,000 among the two types .....

    A trust fund has ₹30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide ₹30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

    (a) ₹1800 (b) ₹2000

    Solution: To solve this problem, we can set up two equations based on the given information and then use matrix multiplication to find the solution.

    Let's denote the amount invested in the first bond as 𝑥x and the amount invested in the second bond as 𝑦y. Then, we have two conditions:

    1. The total amount invested is ₹30,000: 𝑥+𝑦=30000x+y=30000

    2. The total annual interest is either ₹1800 or ₹2000. The interest from the first bond is 5% of 𝑥x, and from the second bond is 7% of 𝑦y. So we have two possible equations depending on the total interest required:

       • For ₹1800 total interest: 0.05𝑥+0.07𝑦=18000.05x+0.07y=1800
       • For ₹2000 total interest: 0.05𝑥+0.07𝑦=20000.05x+0.07y=2000

    Now, let's solve these equations using matrix multiplication.

    Solution for (a) ₹1800 interest:

    The equations can be rewritten in matrix form as:

    [110.050.07][𝑥𝑦]=[300001800][10.05​10.07​][xy​]=[300001800​]

    To find [𝑥𝑦][xy​], we need to multiply the inverse of the coefficient matrix by the matrix of total amounts. Let's compute this:

    First, we find the inverse of the coefficient matrix:

    𝐴=[110.050.07]A=[10.05​10.07​]

    We calculate the inverse of 𝐴A, denoted as 𝐴−1A−1, and then we multiply 𝐴−1A−1 by the total amounts matrix to find [𝑥𝑦][xy​].

    Now, will calculate the inverse and perform the matrix multiplication to solve for 𝑥x and 𝑦y. Let's get started with the calculations.

    For the case where the trust fund must obtain an annual total interest of ₹1800, the amount to be invested in the two types of bonds is:

    • ₹15,000 in the first bond (which pays 5% interest per year).
    • ₹15,000 in the second bond (which pays 7% interest per year).

    Now let's move on to the next part and solve for the case where the total interest required is ₹2000.

    We will use the same coefficient matrix, but with the total amounts matrix adjusted for the ₹2000 interest:

    [110.050.07][𝑥𝑦]=[300002000][10.05​10.07​][xy​]=[300002000​]

    We will calculate this now. ​

    For the case where the trust fund must obtain an annual total interest of ₹2000, the amount to be invested in the two types of bonds is:

    • ₹5,000 in the first bond (which pays 5% interest per year).
    • ₹25,000 in the second bond (which pays 7% interest per year).

    These are the divisions of the ₹30,000 among the two bonds to meet the respective annual interest targets. ​

16. 16.If 𝑛 = 𝑝 n=p, then the order of the matrix 7 𝑋 − 5 𝑍 is: (A) 𝑝 × 2 (B) 2 × 𝑛 (C) 𝑛 × 3 (D) 𝑝 × 𝑛

    If 𝑛=𝑝n=p, then the order of the matrix 7𝑋−5𝑍7X−5Z is:

    (A) 𝑝×2p×2 (B) 2×𝑛2×n (C) 𝑛×3n×3 (D) 𝑝×𝑛p×n

    Solution:

    To find the order of the matrix 7𝑋−5𝑍7X−5Z, we need to know the order of the matrices 𝑋X and 𝑍Z. Since the matrices are being subtracted, they must have the same order. Given that 𝑛=𝑝n=p, the matrices 𝑋X and 𝑍Z must both be square matrices of order 𝑛×𝑛n×n or 𝑝×𝑝p×p since 𝑛=𝑝n=p.

    Therefore, the order of the matrix 7𝑋−5𝑍7X−5Z would also be 𝑛×𝑛n×n or 𝑝×𝑝p×p since both 𝑋X and 𝑍Z have the same order.

    So, the correct answer is:

    (D) 𝑝×𝑛p×n

    Since 𝑛n and 𝑝p are equal, 𝑝×𝑛p×n is the same as 𝑝×𝑝p×p or 𝑛×𝑛n×n.

17. 17.Transpose of a Matrix

    Definition of Transpose of a Matrix

    The transpose of a matrix is a new matrix that's created by swapping the rows and columns of the original matrix. In simpler terms, the element at the 𝑖i-th row and 𝑗j-th column of the original matrix becomes the element at the 𝑗j-th row and 𝑖i-th column in the transposed matrix.

    Example to Illustrate

    Let's consider a real-life example to understand this concept better: Imagine you have a table recording the amount of different fruits (like apples, bananas, and oranges) bought on different days of the week. If your original matrix has days as rows and fruits as columns, the transpose of this matrix would have fruits as rows and days as columns. This flip can help you quickly see which fruit was bought most on each day instead of which day had the most of a particular fruit.

    Mathematical Example

    Suppose we have a matrix 𝐴A:

    𝐴=(123456)A=⎝⎛​135​246​⎠⎞​

    To find the transpose of 𝐴A (denoted as 𝐴𝑇AT), we swap the rows and columns:

    𝐴𝑇=(135246)AT=(12​34​56​)

    In the original matrix 𝐴A, the element "2" is in the first row, second column. In 𝐴𝑇AT, it moves to the second row, first column.

    Practical Application

    In real-world scenarios, the concept of matrix transpose is used in various fields such as computer graphics, where it helps in converting different coordinate systems, and in statistics, where it's used to calculate covariance matrices from data samples.

    Careers and Industries

    Many careers use the concept of matrix transposition, especially:

    1. Data Science: Transposing matrices to reformat data for analysis.
    2. Computer Science: Operations on matrices for algorithms and graphics.
    3. Engineering: For calculations in signal processing and control systems.

    By understanding matrix transposition, you can enhance your skills in handling complex data transformations in these fields.

18. 18.Properties of Transpose of the Matrices

    Properties of the Transpose of Matrices
    

    Understanding the properties of the transpose of matrices can help you solve various mathematical problems more efficiently, particularly in areas like algebra and geometry. Let's explore these properties with examples from everyday life and simple mathematical illustrations.

    Property 1: Transpose of a Transpose

    Property: (𝐴𝑇)𝑇=𝐴(AT)T=A

    Explanation: When you transpose a matrix twice, you get back the original matrix. Imagine flipping a stack of cards horizontally and then doing it again; the cards end up in their original orientation.

    Example:

    𝐴=(1234)A=(13​24​)𝐴𝑇=(1324)AT=(12​34​)(𝐴𝑇)𝑇=(1234)(AT)T=(13​24​)

    Property 2: Transpose of a Sum

    Property: (𝐴+𝐵)𝑇=𝐴𝑇+𝐵𝑇(A+B)T=AT+BT

    Explanation: The transpose of the sum of two matrices is the same as the sum of their transposes. This is like saying swapping the roles of two teams in a relay race (runner positions) before and after team formation leads to the same race setup.

    Example:

    𝐴=(1234),𝐵=(5678)A=(13​24​),B=(57​68​)(𝐴+𝐵)𝑇=(610812),𝐴𝑇+𝐵𝑇=(610812)(A+B)T=(68​1012​),AT+BT=(68​1012​)

    Property 3: Transpose of a Product

    Property: (𝐴𝐵)𝑇=𝐵𝑇𝐴𝑇(AB)T=BTAT

    Explanation: When you transpose a product of two matrices, the order of the matrices is reversed in the transposition. This is akin to applying two filters to a photo in reverse order; the final effect is determined by the order of application.

    Example:

    𝐴=(1234),𝐵=(5678)A=(13​24​),B=(57​68​)𝐴𝐵=(19224350)AB=(1943​2250​)(𝐴𝐵)𝑇=(19432250),𝐵𝑇𝐴𝑇=(19432250)(AB)T=(1922​4350​),BTAT=(1922​4350​)

    Real-World Application

    These properties are particularly useful in fields like computer graphics, where transformations need to be reversed or combined efficiently, and in statistical data analysis, where data structures are often transformed for better accessibility or performance.

    Careers and Industries

    Understanding and applying these properties of matrix transposition can be beneficial in:

    1. Data Science: For data manipulation and machine learning algorithms.
    2. Economics: In econometric models to handle large datasets.
    3. Engineering: For solving systems of equations and optimizing designs.

    By grasping these properties, you can handle various mathematical and practical tasks more effectively in these fields.

19. 19.Symmetric and Skew-Symmetric Matrices

    Symmetric and Skew-Symmetric Matrices

    Understanding symmetric and skew-symmetric matrices is quite valuable, especially when dealing with algebraic problems or understanding geometric transformations. Let’s explore these concepts with practical examples and mathematical definitions.

    Symmetric Matrices

    Definition: A matrix is called symmetric if it is equal to its transpose. That is, a matrix 𝐴A is symmetric if 𝐴=𝐴𝑇A=AT.

    Real-life Example: Imagine a matrix that represents distances between cities, where each entry shows the distance from one city to another. The distance from city A to city B is the same as from city B to city A, making this distance matrix symmetric.

    Mathematical Example:

    𝐴=(123256369)A=⎝⎛​123​256​369​⎠⎞​

    Here, 𝐴=𝐴𝑇A=AT, so matrix 𝐴A is symmetric.

    Applications:

    • Symmetric matrices are extensively used in physics, particularly in theories related to springs and forces, where forces acting in one direction have equal and opposite reactions.
    • In computer science, these matrices help in constructing algorithms for network flow and image processing where symmetry is a common property.

    Skew-Symmetric Matrices

    Definition: A matrix is called skew-symmetric if its transpose is equal to the negative of itself. That is, a matrix 𝐴A is skew-symmetric if 𝐴𝑇=−𝐴AT=−A. Importantly, all the diagonal entries in a skew-symmetric matrix must be zero, because each element must equal its own negative.

    Real-life Example: Consider the angular velocities in mechanics, where the motion in one direction has an equal and opposite effect in another, represented in skew-symmetric matrices.

    Mathematical Example:

    𝐴=(02−1−2031−30)A=⎝⎛​0−21​20−3​−130​⎠⎞​

    Here, 𝐴𝑇=−𝐴AT=−A, so matrix 𝐴A is skew-symmetric.

    Applications:

    • Skew-symmetric matrices are crucial in areas of mechanics and physics dealing with rotational movements.
    • In robotics and aerospace engineering, they are used to describe the relationships between different parts of a robotic arm or an aircraft during flight.

    Careers and Industries

    Symmetric Matrices:

    • Engineering: Used for structural analysis and system dynamics.
    • Economics: Applied in portfolio theory and risk management.

    Skew-Symmetric Matrices:

    • Mechanical Engineering: Used in analyzing stress and rotational forces.
    • Robotics: Helpful in modeling and controlling robot movements.

    By understanding symmetric and skew-symmetric matrices, you can enhance your ability to solve complex problems in various scientific and engineering fields, leading to more efficient and innovative solutions.

20. 20.Theorems of Symmetric and Skew-Symmetric Matrices

    Theorem 1: Every Square Matrix as a Sum of Symmetric and Skew-Symmetric Matrices

    Theorem: Every square matrix 𝐴A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

    Formula:

    𝐴=𝐴+𝐴𝑇2+𝐴−𝐴𝑇2A=2A+AT​+2A−AT​

    Explanation: The first term, 𝐴+𝐴𝑇22A+AT​, is symmetric because when you transpose it, you get the same matrix back. The second term, 𝐴−𝐴𝑇22A−AT​, is skew-symmetric because its transpose gives the negative of itself.

    Example: Consider the matrix

    𝐴=(01−10)A=(0−1​10​)

    Let's break it down:

    Symmetric Part=𝐴+𝐴𝑇2=(01−10)+(0−110)2=(0000)Symmetric Part=2A+AT​=2(0−1​10​)+(01​−10​)​=(00​00​)Skew-Symmetric Part=𝐴−𝐴𝑇2=(01−10)−(0−110)2=(01−10)Skew-Symmetric Part=2A−AT​=2(0−1​10​)−(01​−10​)​=(0−1​10​)

    So, 𝐴A itself is skew-symmetric in this case.

    Theorem 2: Diagonal Elements of a Skew-Symmetric Matrix

    Theorem: All diagonal elements of a skew-symmetric matrix are zero.

    Explanation: For any matrix 𝐴A that is skew-symmetric, the elements on the diagonal must satisfy 𝑎𝑖𝑖=−𝑎𝑖𝑖aii​=−aii​. The only number that is equal to its negative is zero.

    Example: The matrix

    𝐴=(02−1−2031−30)A=⎝⎛​0−21​20−3​−130​⎠⎞​

    clearly shows zero diagonal elements, which confirms its skew-symmetry.

    Theorem 3: Eigenvalues of Symmetric and Skew-Symmetric Matrices

    Theorem: The eigenvalues of a symmetric matrix are always real, and the eigenvalues of a skew-symmetric matrix are always pure imaginary or zero.

    Explanation: Symmetric matrices have real eigenvalues because they can be diagonalized through orthogonal transformations, which do not change the nature of their eigenvalues. Skew-symmetric matrices have purely imaginary eigenvalues because their characteristic equation, which is real, leads to imaginary solutions.

    Example: For the symmetric matrix

    𝐵=(2−10−12−10−12)B=⎝⎛​2−10​−12−1​0−12​⎠⎞​

    The eigenvalues can be calculated and verified to be real.

    These theorems show how symmetric and skew-symmetric matrices behave under certain mathematical operations and properties. These concepts are not just academic; they're widely used in fields such as physics, engineering, and computer science to solve practical problems related to structural dynamics, signal processing, and more.

21. 21.Exercise Questions

    Given two matrices: 𝐴=(−123579−211)A=⎝⎛​−15−2​271​391​⎠⎞​ 𝐵=(−41−5120131)B=⎝⎛​−411​123​−501​⎠⎞​

    Verify that: (i) (𝐴+𝐵)𝑇=𝐴𝑇+𝐵𝑇(A+B)T=AT+BT (ii) (𝐴−𝐵)𝑇=𝐴𝑇−𝐵𝑇(A−B)T=AT−BT

    Solution: Let's solve these one by one.

    For part (i):

    First, we will compute 𝐴+𝐵A+B, then take the transpose of the result, and separately compute 𝐴𝑇+𝐵𝑇AT+BT to verify the equality.

    For part (ii):

    We will compute 𝐴−𝐵A−B, then take the transpose of the result, and separately compute 𝐴𝑇−𝐵𝑇AT−BT to verify the equality.

    Let's do these calculations.

    The calculations verify the properties as follows:

    For part (i), (𝐴+𝐵)𝑇(A+B)T is: (−56−1394−292)⎝⎛​−53−2​699​−142​⎠⎞​ and 𝐴𝑇+𝐵𝑇AT+BT is: (−56−1394−292)⎝⎛​−53−2​699​−142​⎠⎞​ They are equal, confirming the property that (𝐴+𝐵)𝑇=𝐴𝑇+𝐵𝑇(A+B)T=AT+BT.

    For part (ii), (𝐴−𝐵)𝑇(A−B)T is: (34−315−2890)⎝⎛​318​459​−3−20​⎠⎞​ and 𝐴𝑇−𝐵𝑇AT−BT is: (34−315−2890)⎝⎛​318​459​−3−20​⎠⎞​ They are also equal, confirming the property that (𝐴−𝐵)𝑇=𝐴𝑇−𝐵𝑇(A−B)T=AT−BT.

22. 22.Exercise Questions

    A=[cosα−sinα​sinαcosα​] then verify that A' A = 1
    
    Solution: Let's verify that

    𝐴′𝐴=𝐼A′A=I where 𝐴A is the given matrix:

    𝐴=[cos⁡𝛼sin⁡𝛼−sin⁡𝛼cos⁡𝛼]A=[cosα−sinα​sinαcosα​]

    Step 1: Find the transpose of A, 𝐴′A′ The transpose of a matrix is obtained by swapping its rows and columns. Thus, the transpose 𝐴′A′ of 𝐴A is:

    𝐴′=[cos⁡𝛼−sin⁡𝛼sin⁡𝛼cos⁡𝛼]A′=[cosαsinα​−sinαcosα​]

    Step 2: Multiply 𝐴′A′ with 𝐴A Now, we need to multiply 𝐴′A′ by 𝐴A:

    𝐴′𝐴=[cos⁡𝛼−sin⁡𝛼sin⁡𝛼cos⁡𝛼][cos⁡𝛼sin⁡𝛼−sin⁡𝛼cos⁡𝛼]A′A=[cosαsinα​−sinαcosα​][cosα−sinα​sinαcosα​]

    We calculate the product by matrix multiplication:

    (𝐴′𝐴)11=cos⁡𝛼cos⁡𝛼+(−sin⁡𝛼)(−sin⁡𝛼)=cos⁡2𝛼+sin⁡2𝛼,(𝐴′𝐴)12=cos⁡𝛼sin⁡𝛼+(−sin⁡𝛼)cos⁡𝛼=cos⁡𝛼sin⁡𝛼−sin⁡𝛼cos⁡𝛼=0,(𝐴′𝐴)21=sin⁡𝛼cos⁡𝛼+cos⁡𝛼(−sin⁡𝛼)=sin⁡𝛼cos⁡𝛼−sin⁡𝛼cos⁡𝛼=0,(𝐴′𝐴)22=sin⁡𝛼sin⁡𝛼+cos⁡𝛼cos⁡𝛼=sin⁡2𝛼+cos⁡2𝛼.(A′A)11​(A′A)12​(A′A)21​(A′A)22​​=cosαcosα+(−sinα)(−sinα)=cos2α+sin2α,=cosαsinα+(−sinα)cosα=cosαsinα−sinαcosα=0,=sinαcosα+cosα(−sinα)=sinαcosα−sinαcosα=0,=sinαsinα+cosαcosα=sin2α+cos2α.​

    Since cos⁡2𝛼+sin⁡2𝛼=1cos2α+sin2α=1 (a basic trigonometric identity), the matrix product 𝐴′𝐴A′A simplifies to:

    𝐴′𝐴=[1001]=𝐼A′A=[10​01​]=I

    Thus, 𝐴′𝐴A′A equals the identity matrix 𝐼I, confirming the statement.

    
    𝐴′𝐴=𝐼

23. 23.Show that the matrix 𝐴 is a symmetric matrix.

    Show that the matrix 𝐴A is a symmetric matrix.

    The given matrix is:

    𝐴=[1−15−121513]A=⎣⎡​1−15​−121​513​⎦⎤​

    Solution: To verify if 𝐴A is symmetric, we need to check if 𝐴=𝐴𝑇A=AT, where 𝐴𝑇AT is the transpose of 𝐴A. The transpose is obtained by swapping rows for columns in 𝐴A.

    Thus, 𝐴𝑇AT is:

    𝐴𝑇=[1−15−121513]AT=⎣⎡​1−15​−121​513​⎦⎤​

    Comparing 𝐴A and 𝐴𝑇AT:

    Both 𝐴 and 𝐴𝑇 are identical, hence 𝐴 is symmetric.Both A and AT are identical, hence A is symmetric.

24. 24.Show that the matrix A is a skew symmetric matrix.

    Show that the matrix 𝐴A is a skew symmetric matrix.

    The matrix is:

    𝐴=[01−1−1011−10]A=⎣⎡​0−11​10−1​−110​⎦⎤​

    Solution: A matrix is skew symmetric if 𝐴𝑇=−𝐴AT=−A. We calculate 𝐴𝑇AT:

    𝐴𝑇=[0−1110−1−110]AT=⎣⎡​01−1​−101​1−10​⎦⎤​

    Now, checking 𝐴𝑇=−𝐴AT=−A:

    −𝐴=[−0−111−0−1−11−0]−A=⎣⎡​−01−1​−1−01​1−1−0​⎦⎤​

    Since 𝐴𝑇AT and −𝐴−A are identical:

    Thus, 𝐴 is skew symmetric.Thus, A is skew symmetric.

25. 25.Invertible Matrices

    An invertible matrix is a square matrix that has an inverse. A matrix 𝐴A is invertible if there exists another matrix 𝐵B such that when 𝐴A is multiplied by 𝐵B, the result is the identity matrix 𝐼I. The identity matrix is a special type of matrix where all the elements on the diagonal are 1, and all off-diagonal elements are 0. The matrix 𝐵B is called the inverse of 𝐴A, and it's denoted as 𝐴−1A−1.

    Characteristics of Invertible Matrices:

    1. Square Matrix: Only square matrices (matrices with the same number of rows and columns) can be invertible.
    2. Non-zero Determinant: For a matrix to be invertible, its determinant must be non-zero. If the determinant is zero, the matrix is said to be singular or non-invertible.
    3. Unique Inverse: Each invertible matrix has exactly one inverse.

    How to Determine if a Matrix is Invertible:

    • Compute the determinant of the matrix. If the determinant is not zero, the matrix is invertible.
    • Attempt to find the inverse through methods like Gaussian elimination or the adjugate matrix method. If these methods yield a matrix, the original matrix is invertible.

    How to Calculate the Inverse:

    For a 2x2 matrix 𝐴=[𝑎𝑏𝑐𝑑]A=[ac​bd​]:

    • The inverse is given by:𝐴−1=1𝑎𝑑−𝑏𝑐[𝑑−𝑏−𝑐𝑎]A−1=ad−bc1​[d−c​−ba​]provided that 𝑎𝑑−𝑏𝑐≠0ad−bc=0 (i.e., the determinant is non-zero).

    For larger matrices, the inverse might be computed using:

    • Gaussian Elimination: Transforming the matrix into its reduced row echelon form.
    • Adjugate Method: Computing the adjugate (or adjoint) matrix and dividing each element by the determinant of the original matrix.

    Real-World Applications:

    • Cryptography: Invertible matrices are used in coding and encryption algorithms to encode messages securely.
    • Economics and Statistics: Invertible matrices are crucial in solving systems of linear equations, which is common in economic modeling and statistical analysis.
    • Engineering: Used in solving equations in electrical circuits, structural analysis, and control systems.

    Example:

    Suppose we have a matrix 𝐴=[1234]A=[13​24​]. To find if 𝐴A is invertible and find its inverse:

    1. Calculate the determinant: det(𝐴)=1(4)−2(3)=4−6=−2det(A)=1(4)−2(3)=4−6=−2.
    2. Since the determinant is not zero, 𝐴A is invertible.
    3. The inverse of 𝐴A is:𝐴−1=1−2[4−2−31]=[−211.5−0.5]A−1=−21​[4−3​−21​]=[−21.5​1−0.5​]

26. 26.Theorems of Invertible Matrix

    Theorem 1: Invertible Matrix Theorem

    This theorem provides a list of equivalent conditions, meaning if one is true, all are true. Let's explore some of these conditions with a numerical example:

    Matrix A: 𝐴=[1234]A=[13​24​]

    1. 𝐴A is invertible.
    2. The determinant of 𝐴A is not zero (det⁡(𝐴)≠0det(A)=0).
       • Determinant Calculation: det⁡(𝐴)=1(4)−2(3)=4−6=−2det(A)=1(4)−2(3)=4−6=−2. Since the determinant is non-zero, 𝐴A is invertible.
    3. There is a matrix 𝐵B such that 𝐴𝐵=𝐼AB=I and 𝐵𝐴=𝐼BA=I.
       • Finding Inverse 𝐴−1A−1: 𝐴−1=1det⁡(𝐴)[4−2−31]=1−2[4−2−31]=[−211.5−0.5]A−1=det(A)1​[4−3​−21​]=−21​[4−3​−21​]=[−21.5​1−0.5​]
       • Verification: 𝐴𝐴−1=[1001]AA−1=[10​01​]

    Theorem 2: The Product of Invertible Matrices

    If 𝐴A and 𝐵B are both invertible matrices, their product 𝐴𝐵AB is also invertible, and the inverse is (𝐴𝐵)−1=𝐵−1𝐴−1(AB)−1=B−1A−1.

    Example: Let's consider 𝐴A from above and 𝐵=[0110]B=[01​10​].

    • 𝐵B is invertible with 𝐵−1=𝐵B−1=B (because 𝐵𝐵=𝐼BB=I).
    • Compute 𝐴𝐵AB and find (𝐴𝐵)−1(AB)−1: 𝐴𝐵=[1234][0110]=[2143]AB=[13​24​][01​10​]=[24​13​] (𝐴𝐵)−1=𝐵−1𝐴−1=[0110][−211.5−0.5]=[1.5−0.5−21](AB)−1=B−1A−1=[01​10​][−21.5​1−0.5​]=[1.5−2​−0.51​]

    Theorem 3: The Inverse of a Transpose

    The transpose of an invertible matrix is also invertible, and the inverse of the transpose is the transpose of the inverse: (𝐴𝑇)−1=(𝐴−1)𝑇(AT)−1=(A−1)T

    Example: Using 𝐴A from earlier,

    • 𝐴𝑇=[1324]AT=[12​34​]
    • (𝐴−1)𝑇=[−21.51−0.5](A−1)T=[−21​1.5−0.5​]
    • Verification shows 𝐴𝑇(𝐴−1)𝑇=𝐼AT(A−1)T=I

    Real-World Application

    These theorems are used in various fields:

    • Cryptography to secure data by encoding and decoding it using invertible matrices.
    • Economic modeling where systems of equations based on invertible matrices predict market behaviors.
    • Engineering for designing systems that rely on matrix operations to model behaviors and solve equations.

    These examples and applications show how theoretical matrix concepts apply to solving real-world problems, enhancing our understanding and providing tools for practical application.

27. 27.Matrices A and B will be inverse of each other only if (A) AB = BA (B) AB = BA = 0 (C) AB = 0, BA = I (D) AB = BA = I

    Statements Analysis:

    • Statement (A): 𝐴𝐵=𝐵𝐴AB=BA
    • Statement (B): 𝐴𝐵=𝐵𝐴=0AB=BA=0
    • Statement (C): 𝐴𝐵=0,𝐵𝐴=𝐼AB=0,BA=I and (𝐷)𝐴𝐵=𝐵𝐴=𝐼(D)AB=BA=I

    Correct Understanding:

    Matrices 𝐴A and 𝐵B are inverses of each other if 𝐴𝐵=𝐵𝐴=𝐼AB=BA=I, where 𝐼I is the identity matrix. This means:

    1. 𝐴𝐵=𝐼AB=I - This shows that 𝐵B is a right-inverse of 𝐴A.
    2. 𝐵𝐴=𝐼BA=I - This shows that 𝐵B is a left-inverse of 𝐴A.

    For square matrices, if 𝐴A has both a left-inverse and a right-inverse, they must be the same, and 𝐴A is invertible.

    Analysis of Given Statements:

    • Statement (A): Simply stating 𝐴𝐵=𝐵𝐴AB=BA does not necessarily imply that 𝐴𝐵AB or 𝐵𝐴BA equals the identity matrix 𝐼I. They could be equal and non-identity matrices.
    • Statement (B): 𝐴𝐵=𝐵𝐴=0AB=BA=0 suggests that both products are the zero matrix, which is incorrect because it contradicts the definition of inverse matrices.
    • Statement (C): 𝐴𝐵=0,𝐵𝐴=𝐼AB=0,BA=I is inconsistent. If 𝐵𝐴=𝐼BA=I, then 𝐴A and 𝐵B cannot yield 𝐴𝐵=0AB=0 if both are non-zero matrices, as that would violate properties of the identity matrix and invertibility.
    • Statement (D): 𝐴𝐵=𝐵𝐴=𝐼AB=BA=I correctly states that 𝐴A and 𝐵B are inverses of each other, as both products are the identity matrix.

    Conclusion:

    The only correct statement that accurately describes the condition under which matrices 𝐴A and 𝐵B are inverses of each other is Statement (D): 𝐴𝐵=𝐵𝐴=𝐼AB=BA=I. This condition confirms that 𝐴A and 𝐵B are inverse matrices, fulfilling the requirement for each to undo the operation of the other, thus returning the identity matrix.