Matrix Inverse

The multiplication of a matrix by its inverse is equal to the identity matrix.

A · A-1  = A-1 · A = I

Properties of the Inverse Matrix

(A · B)-1  = B-1 · A-1

(A-1)-1  = A

(k · A)-1  = k-1 · A-1

(At)-1  = (A-1)t


Steps to Calculate the Inverse Matrix

A is a square matrix of order n. To calculate the inverse of A, denoted as A-1, follow these steps:

1 Construct a matrix of type M = (A | I), that is to say, A is in the left half of M and the identity matrix I is on the right.

Consider an arbitrary 3x3 matrix:

Matrix A

Place the identity matrix of order 3 to the right of Martix M.

Identity Matrix

2Using the Gaussian elimination method, transform the left half, A, to the identity matrix, located to the right, and the matix that results in the right side will be the inverse of matrix: A-1.

r2 - r1

Gaussian Elimination Method

r3 + r2

Gaussian Elimination Method

r2 - r3

Gaussian Elimination Method

r1 + r2

Gaussian Elimination Method

(−1) · r2

Gaussian Elimination Method

The inverse matrix is:

>Inverse Matrix

Examples

Calculate the matrix inverse of:

Matrix

1 Construct a matrix of type M = (A | I).

Inverse of a Matrix

2Using the Gaussian elimination method, transform the left half, A, in the identity matrix, located to the right, and the matix that results in the right side will be the inverse of matrix: A-1.

Inverse of a Matrix Operations

Inverse of a Matrix Operations

Inverse of a Matrix Operations

Inverse of a Matrix


For what values of m in the matrix    Matrix    does not support an inverse?

Matrix Inverse Operations

Matrix Inverse Solution

For any real value of m, there is the inverse A-1.


Calculating the matrix inverse for determinats