Inverse finding using elementary row operations

inverse finding

1. Linear Algebra Determinants Invertible Matrices Inverse by Reduce Row echelon Form Determinant Properties 4/16/2021 Linear Algebra 1

2. Determinant If then , Determinant is a number and is represented by . A determinant is used to compute the inverse of a Matrix. Such as: If then the matrix is not invertible and is called Singular. If then the matrix is invertible and it is said to be non-singular. Adjoint: Adjoint of a Matrix is denoted by , if then is its Adjoint, but what if we have a 3*3 matrix or bigger than that? Then to find Adjoint of such a matrix we compute Minors & Cofactors.   4/16/2021 Linear Algebra 2

3. Minors & Cofactors Minors: If is a square matrix then the Minor of entry is denoted by and is defined to be the determinant of the sub-matrix that remains after the row and column are deleted from . Cofactor: The number is denoted by and is called the Cofactor of entry . Example: If & & & & matrix of cofactors A= , det (A)=4(0)-2(12-0)+(9)=-15 & & & &   4/16/2021 Linear Algebra 3 &  

4. Adjoint of a Matrix The Adjoint of a matrix is the transpose of Matrix of cofactors, therefore:   4/16/2021 Linear Algebra 4

5. Inverse of a Matrix Let is a matrix, then the minors and cofactors are:   4/16/2021 Linear Algebra 5 Therefor, The cofactors matrix is:   Since, the Adjoint is transpose of cofactors matrix, this yields  

6. Cont. Determinant of , so the Inverse of is: So   4/16/2021 Linear Algebra 6

7. Elementary Matrix An matrix is called an elementary matrix when it can be obtained from the identity matrix by a single elementary row operation. Examples: Elementary matrices are very useful because they enable us to use matrix multiplication to perform elementary row operations. e.g.   4/16/2021 Linear Algebra 7

8. Examples: Elementary Matrices and Elementary Row Operations 4/16/2021 Linear Algebra 8

9. Elementary Matrices are Invertible If is an elementary matrix, then exists and is invertible. A square matrix is invertible if and only if it can be written as the product of Elementary matrices. The determinant of an elementary matrix can not be zero.   4/16/2021 Linear Algebra 9

10. Properties of Determinant If and are square matrices of order , then If is a square matrix of order and is a scalar, then the is: A square matrix is invertible if and only if If is an matrix then If is a square matrix then If one row of a matrix consists entirely of zeros, then the determinant is zero. If two rows of a matrix are interchanged, the determinant changes sign. If two rows of a matrix are identical, the determinant is zero . The determinant of a triangular matrix, either upper or lower, is the product of the elements on the main diagonal.   4/16/2021 Linear Algebra 10

11. Inverse of a Matrix using Row Operations If is an matrix, then we can easily find its inverse by elementary row operations, by: Then solving it using row operations, we’ll have the Inverse. e.g. Find Inverse of: Solution:   4/16/2021 Linear Algebra 11

12. Example Cont. 4/16/2021 Linear Algebra 12

13. 4/16/2021 Linear Algebra 13