Matrices Upto Inverse PDF

Matrices and their Properties 19-09-2024 School of Basic Sciences 1  Outline Revision of Matrix Types of Matrices Elementary Row Operations on Matrices Echelon form of a Matrix The rank of a Matrix The inverse of a Matrix using the Gauss Jordan M...

Matrices and their Properties 19-09-2024 School of Basic Sciences 1  Outline Revision of Matrix Types of Matrices Elementary Row Operations on Matrices Echelon form of a Matrix The rank of a Matrix The inverse of a Matrix using the Gauss Jordan Method 19-09-2024 School of Basic Sciences 2  Revision  A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. 1 2 3 1 2 3 4 1 2 Example: A = , B= 4 5 6 , C= 2 4 6 8 3 4 7 8 9 3 6 9 12  The order of a matrix (or dimension of a matrix) refers to the number of rows and columns in the matrix. If a matrix has m rows and n columns then the order of the matrix is 𝑚 × 𝑛 and it is written as Am×n. Example: In the above-defined matrix A2×2 , B3×3 and C3×4. 19-09-2024 School of Basic Sciences 3  Revision  It is denoted by a capital letter of the alphabet say, 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛 , where 1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛, 𝑚 is the number of rows and 𝑛 is the number of columns. In particular, 𝑎23 entry in any matrix refers to the 3rd element in the 2nd row. 1 2 3 4 Example: In matrix C = 2 4 6 8 , 𝑎23 = 6; 𝑎33 = 9. 3 6 9 12 19-09-2024 School of Basic Sciences 4  Types of Matrices  A matrix is said to Null (Zero) Matrix if all the entries of the matrix are zero. 0 0 0 0 0 0 0 0 0 Example: A2×2 = , B3×3 = 0 0 0 , C3×4 = 0 0 0 0 0 0 0 0 0 0 0 0 0  A matrix is a square matrix if the number of rows is equal to the number of columns. 1 2 3 1 2 Example: A2×2 = , B3×3 = 4 5 6 3 4 7 8 9 19-09-2024 School of Basic Sciences 5  Types of Matrices  A diagonal matrix of order 𝑛 is a type of square matrix in which all the elements outside the main diagonal are zero. Formally, a diagonal matrix 𝑎11 ⋯ 0 𝐷= ⋮ ⋱ ⋮. Here, 𝑎𝑖𝑖 are the diagonal entries. 0 ⋯ 𝑎𝑛𝑛 𝑛×𝑛  An identity matrix is a special type of square matrix where all the diagonal elements are 1, and all the off-diagonal elements are 0. It is denoted by 𝐼𝑛 for an 𝑛 × 𝑛 matrix. 1 0 0 1 0 Example: I2×2 = , I3×3 = 0 1 0 0 1 0 0 1 19-09-2024 School of Basic Sciences 6  Types of Matrices  A scalar matrix of order 𝑛 is a type of square matrix in which all the elements outside the main diagonal are zero and each diagonal entry is the same. 𝑘 0 0 2 0 Example: A2×2 = , B3×3 = 0 𝑘 0 = 𝑘𝐼 0 2 0 0 𝑘  The trace of a matrix is the sum of the elements on the main diagonal of a square matrix. It is denoted by 𝑇𝑟 𝐴 = 𝑎11 + 𝑎22 + ⋯ + 𝑎𝑛𝑛. 1 2 3 1 2 Example: A = , Tr(A)=1+4=5; B= 4 5 6 , Tr(B)=15. 3 4 7 8 9 19-09-2024 School of Basic Sciences 7  Operations on Matrices 1) Matrix Addition 4) Scalar Multiplication 𝐴 = 𝑎𝑖𝑗 , 𝐵 = 𝑏𝑖𝑗 , 𝐴 + 𝐵 = 𝑎𝑖𝑗 + 𝑏𝑖𝑗 𝐴 = 𝑎𝑖𝑗 , 𝑐𝐴 = 𝑐𝑎𝑖𝑗 𝑚×𝑛 𝑚×𝑛 𝑚×𝑛 𝑚×𝑛 𝑚×𝑛 2) Matrix Subtraction 5) Transpose of a Matrix 𝐴 = 𝑎𝑖𝑗 , 𝐵 = 𝑏𝑖𝑗 , 𝐴 − 𝐵 = 𝑎𝑖𝑗 − 𝑏𝑖𝑗 𝐴 = 𝑎𝑖𝑗 , 𝐴𝑇 = 𝑎𝑖𝑗 𝑚×𝑛 𝑚×𝑛 𝑚×𝑛 𝑚×𝑛 𝑛×𝑚 3) Matrix Multiplication 𝑛 𝐴 = 𝑎𝑖𝑗 , 𝐵 = 𝑏𝑖𝑗 , 𝐴. 𝐵 = 𝑗=1 𝑎𝑖𝑗 𝑏𝑗𝑘 𝑚×𝑝 𝑚×𝑛 𝑞×𝑝 exits only if 𝑛 = 𝑞 19-09-2024 School of Basic Sciences 8  Activity-1 1. Compute the product of the matrices 1 2 5 6 𝐴= ,𝐵=. Verify 3 4 7 8 whether matrix multiplication is commutative by checking if 𝐴. 𝐵 = 𝐵. 𝐴. 2. If 𝐴 and 𝐵 are square matrix of same order, is it always true that 𝐴. 𝐵 = 𝐵. 𝐴? Justify your answer with an example. 19-09-2024 School of Basic Sciences 9  Determinant of Matrices The determinant of a square matrix is a scalar value which is primarily used to determine whether a matrix is invertible, to solve systems of linear equations, and in various applications of linear algebra. 1) Determinant of a 2 × 2 Matrix: 𝑎 𝑏 𝐴= ⇒ det 𝐴 = 𝑎𝑑 − 𝑏𝑐 𝑐 𝑑 19-09-2024 School of Basic Sciences 10  Determinant of Matrices 2) Determinant of a 3 × 3 Matrix: 𝑎 𝑏 𝑐 𝐴= 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 ⇒ det 𝐴 = 𝑎 𝑒𝑖 − 𝑓ℎ − 𝑏 𝑑𝑖 − 𝑓𝑔 + 𝑐(𝑑ℎ − 𝑒𝑔) 19-09-2024 School of Basic Sciences 11 Properties of Determinant 1) A matrix is invertible if and only if its determinant is non-zero. 2) Certain row operations affect the determinant: − Swapping two rows multiplies the determinant by −1. − Multiplying a row by a scalar multiplies the determinant by that scalar. − Adding a multiple of one row to another does not change the determinant. 3) The determinant of a matrix is the same as the determinant of its transpose. 4) The determinant of the product of two matrices is the product of their determinants: det 𝐴𝐵 = det 𝐴. det(𝐵) 5) If all the elements of a row (or column) are zero, then the determinant is zero. 19-09-2024 School of Basic Sciences 12 Question Time: Can matrices be used in real-life applications? 19-09-2024 School of Basic Sciences 13 Learning outcomes By the end of this topic students will be able to:  Understand elementary row operations.  Apply elementary row operations to convert a matrix into a simpler form.  Calculate the rank of a matrix.  Use the Gauss-Jordan method to find the inverse of a matrix. 19-09-2024 School of Basic Sciences 14 Row Echelon Form (REF) or Echelon Form of a Matrix a) All the zero rows of the matrix are in bottom. 2 2 3 2 2 3 Example: 0 1 0 , 0 0 0 0 0 0 0 1 0 b) Each leading in a column is the right side of the leading column in the previous row. 2 2 3 2 2 3 Example: 0 1 0 , 0 0 1 0 0 0 0 1 0 The matrix which satisfies the above conditions is called in Echelon form. 2 2 3 Example: 0 1 0 is in Echelon form 0 0 0 19-09-2024 School of Basic Sciences 15  Elementary Transformations 1) Interchanging any two rows(or columns) of the matrix. If 𝑅i and 𝑅j are interchanged then it is denoted as 𝑅i ↔ 𝑅j 2 2 3 0 1 0 Example: A= 0 1 0 , after 𝑅1 ↔ 𝑅2 we get 2 2 3 0 0 0 0 0 0 2) Multiplication of any row of matrix by a nonzero number k. It is denoted as 𝑅i ← k𝑅i 6 6 9 Example: In the above example we multiply 𝑅1 by 3 we get 0 1 0 0 0 0 3) Addition of constant multiplication of the elements of any row 𝑅i to the corresponding elements of any other row 𝑅j is denoted by 𝑅i +k𝑅j 2 4 3 Example: After applying 𝑅1 → 𝑅1 + 2𝑅2 in the above example we get 0 1 0 0 0 0 19-09-2024 School of Basic Sciences 16 Row Reduced Echelon Form(RREF) of a Matrix This is a special form of REF. Every matrix has a unique RREF. A matrix is in row reduced echelon form when it satisfies the following conditions: a) It is in REF. b) The first (or leading or pivot) non-zero element in a non-zero row is 1. 1 2 3 2 2 3 1 2 3 Example: 0 1 0 , 0 0 0 , 0 0 1 0 0 0 0 1 0 0 1 0 19-09-2024 School of Basic Sciences 17  Example 1 2 3 Reduce the following matrix: 𝐴 = 4 5 6 into row echelon form. 7 8 9 Solution: Steps to Convert to REF: 1) Pivot in the First Row: The element in the first row and the first column is already 1 (pivot). Zero out elements below this pivot. Apply Row Elementary operations, 1 2 3 1 2 3 1 2 3 4 5 6 𝑅2 → (𝑅2 −4𝑅1 )~ 0 −3 −6 𝑅3 → (𝑅3 −7𝑅1 ) ~ 0 −3 −6 7 8 9 7 8 9 0 −6 −12 19-09-2024 School of Basic Sciences 18  Example 2) Pivot in the Second Row: Make the element in the second row, second column a leading 1 by dividing the entire row by -3. 1 2 3 1 𝑅2 → 𝑅 , ~ 0 1 2 −3 2 0 −6 −12 3) Zero out Elements Below Pivot: Add 6 times the second row to the third row. 1 2 3 𝑅3 → 𝑅3 + 6𝑅2 , ~ 0 1 2 0 0 0 The matrix is now in Row Echelon Form (REF). 19-09-2024 School of Basic Sciences 19  Row Echelon Form Note: − Row Echelon Form need not be unique. − Every matrix has a unique RREF. − The process of converting a matrix to Echelon form is known as the Gaussian Elimination process. 19-09-2024 School of Basic Sciences 20  Rank of a Matrix Rank of a Matrix The rank of a matrix is the maximum number of non-zero rows or columns of the matrix in Row Echelon form. Calculating the Rank To find the rank of a matrix, you can transform it into its Row Echelon Form (REF). After transforming into REF, the Rank of the matrix will be number of non-zero rows in the matrix. 19-09-2024 School of Basic Sciences 21  Rank of a Matrix 1 2 3 Example: Find the Rank of 𝐴 = 4 5 6. 7 8 9 Solution: We have already reduced this matrix into row echelon form, which is given by, 1 2 3 1 2 3 4 5 6 ~ 0 1 2 7 8 9 0 0 0 Since the number of non-zero rows in row echelon form is 2, therefore rank(A)=2 Properties of a Rank: 1) For an 𝑚 × 𝑛 matrix, the rank is a non-negative integer that is at most min(𝑚, 𝑛). It cannot exceed the number of rows or columns. 2) For a square matrix 𝐴 (i.e., 𝑚 = 𝑛), if the rank is 𝑛, then 𝐴 is invertible. 3) The rank of a zero matrix is 0, as it has no non-zero rows or columns. 19-09-2024 School of Basic Sciences 22 Activity-3 (Group Activity on Wooclap) Ques: Find the rank of the following matrices: 1 1 1 1 3 −3 (a) (b) 2 2 2 3 9 −4 3 3 3 1 2 3 4 (c) 2 4 6 8 3 6 9 12 19-09-2024 School of Basic Sciences 23  Inverse of a Matrix Let 𝐴 be a square matrix of order 𝑛 × 𝑛. 𝐴 is said to be an invertible matrix if there exists a matrix 𝐵 of size 𝑛 × 𝑛 such that 𝐴 ∗ 𝐵 = 𝐵 ∗ 𝐴 = 𝐼 The matrix A is called the inverse of B and B is the inverse of A. Conditions for Inverse: 1) The matrix must be square (same number of rows and columns). 2) The determinant of the matrix must be non-zero (det(𝐴) ≠ 0). Properties: 1) (𝐴−1 )−1 = 𝐴 2) (𝐴𝐵)−1 = 𝐵−1 𝐴−1 −1 1 −1 3)(𝐴𝑇 )−1 = (𝐴−1 )𝑇 4) (𝑐𝐴) = 𝐴 𝑐 5) 𝐼 −1 = 𝐼 19-09-2024 School of Basic Sciences 24  Finding inverse of a Matrix Methods for finding the Inverse of the matrix using Gauss-Jordan Elimination method: For a square matrix 𝐴, 1) Set up the augmented matrix: Form an augmented matrix by placing the identity matrix 𝐼 of the same order as 𝐴 beside it, i.e., [𝐴|𝐼]. 2) Apply row operations: − Perform elementary row operations (row swapping, row scaling, and row addition/subtraction) to transform the left side 𝐴 of the augmented matrix into the identity matrix 𝐼. − Simultaneously apply the same operations to the right side (which starts as the identity matrix). 3) Result: Once the left side becomes the identity matrix, the right side will become the inverse of 𝐴, i.e., [𝐼|𝐴−1 ]. 19-09-2024 School of Basic Sciences 25  Example 2 1 1 Consider the matrix: 𝐴 = 1 3 2. Find the inverse of 𝐴 using Gauss Jordan method. 1 0 0 First to find determinant of A, det 𝐴 = −1 ≠ 0, which means its inverse exist. Now, we’ll apply Gauss Jordan method. 1) Set up the augmented matrix [𝐴|𝐼] 2 1 1 | 1 0 0 1 3 2 | 0 1 0 1 0 0 | 0 0 1 2) Apply row operations 1 0.5 0.5 | 0.5 0 0 𝑅1 𝑅1 → 2 ~ 1 3 2 | 0 1 0 1 0 0 | 0 0 1 19-09-2024 School of Basic Sciences 26  Example 1 0.5 0.5 | 0.5 0 0 𝑅2 → 𝑅2 − 𝑅1 ~ 0 2.5 1.5 | −0.5 1 0 1 0 0 | 0 0 1 1 0.5 0.5 | 0.5 0 0 𝑅3 → 𝑅3 − 𝑅1 ~ 0 2.5 1.5 | −0.5 1 0 0 −0.5 −0.5 | −0.5 0 1 and so on (Try yourself). 1 0 0 | 0 0 1 𝑅1 → 𝑅1 − 0.5 × 𝑅2 ~ 0 1 0 | −2 1 3 0 0 1 | 3 −1 −5 3) The inverse matrix is on the right-hand side: 0 0 1 𝐴−1 = −2 1 3 3 −1 −5 19-09-2024 School of Basic Sciences 27 Activity-3 (Group Activity on Wooclap) Ques: Find the rank of the following matrices. If the rank is full then find the inverse of the square matrix. 1 2 −1 0 1 3 (a) 2 4 −2 (b) 4 −5 2 −1 −2 1 3 7 −9 1 2 3 1 3 −3 (c) (d) 2 1 4 3 9 −4 3 0 5 1 1 1 1 2 3 4 (e) 2 2 2 (f) 2 4 6 8 3 3 3 3 6 9 12 19-09-2024 School of Basic Sciences 28 Activity-4 (Group Activity) Find the rank and inverse of the following matrices: 2 1 1 2 1 (1) (2) 0 1 3 3 4 4 2 1 2 1 −1 6 3 (3) 1 0 −1 (4) 4 5 1 1 2 3 1 1 6 3 (5) −1 2 1 (6) 6 5 1 1 −1 19-09-2024 School of Basic Sciences 29  Activity-5 (Jigsaw Activity) Given the matrix 1 2 3 4 𝐴= 0 1 2 3 4 3 2 1 2 1 0 −1 Determine if 𝐴 is invertible by calculating the determinant. If it is invertible, use the Gauss-Jordan elimination method to find 𝐴−1. Verify your result using the properties of the inverse of a matrix. 19-09-2024 School of Basic Sciences 30  Summary Add, subtract, and multiply matrices or multiply them by a scalar. A scalar value that indicates if a square matrix is invertible. A non-zero determinant means the matrix is invertible. The number of non-zero rows or columns in a matrix (Echelon form), indicating the matrix's dimension. A matrix 𝐴 has an inverse 𝐴−1 if it’s square and its determinant is non-zero. The inverse satisfies 𝐴. 𝐴−1 = 𝐼. Augment the matrix with the identity matrix and apply row operations until the original matrix is the identity matrix; the augmented part then becomes the inverse. These methods help solve linear systems, analyze matrix properties, and perform matrix-based computations. 19-09-2024 School of Basic Sciences 31  Practice Questions for LMS 2 −1 7 2 1. Given matrices 𝐴 = ,𝐵= , compute: 6 9 8 12 𝐴+𝐵 𝐴−𝐵 𝐴. 𝐵 9 2 2. Calculate the determinant of the matrix, 𝐴 =. 12 8 12 −11 3. Determine the rank of the matrix, 𝐴 = 2 55. 16 0 4. Use the Gauss-Jordan method to find the inverse of the matrix, 𝐴 = 1 12 3 1 0 4. 5 6 0 2 0 5. Let 𝐴 =. Compute 𝐴. 𝐴−1 and verify if it equals the identity 1 3 matrix. 19-09-2024 School of Basic Sciences 32