Inverse of a Matrix (Method 2)

Inverse of a Matrix
Matrix Inverse
Multiplicative Inverse of a Matrix

For a square matrix A, the inverse is written A^-1. When A is multiplied by A^-1 the result is the identity matrix I. Non-square matrices do not have inverses.

Note: Not all square matrices have inverses. A square matrix which has an inverse is called invertible or nonsingular, and a square matrix without an inverse is called noninvertible or singular.

AA^-1 = A^-1A = I

Example:

For matrix 2x2 matrix A with values: row 1: 4, 3; row 2: 3, 2

, its inverse is

since

AA^-1 =

and A^-1A = .

Here are three ways to find the inverse of a matrix:

1. Shortcut for 2x2 matrices

For , the inverse can be found using this formula:

Example: Matrix inverse example: [1,2; 3,4]⁻¹ = (1/-2)[4,-2; -3,1] = [-2,1; 3/2,-1/2]

2.Augmented matrix method

Use Gauss-Jordan elimination to transform [ A | I ] into [ I | A^-1 ].

Example: The following steps result in Matrix [1 2 / 3 4] raised to the power of -1, representing the inverse of a 2x2 matrix. .

Row reduction steps finding inverse of matrix [[1,2],[3,4]] using augmented matrix, yielding inverse [[-2,1],[3/2,-1/2]]

so we see that Matrix equation showing [1 2; 3 4] inverse equals [-2 1; 3/2 -1/2] .

3. Adjoint method

A^-1 = (adjoint of A) or A^-1 = (cofactor matrix of A)^T

Example: The following steps result in A^-1 for 3×3 matrix A with rows [1,2,3], [0,4,5], [1,0,6], used to illustrate matrix inverse concept. .

The cofactor matrix for A is 3×3 matrix with rows [24, 5, -4], [-12, 3, 2], [-2, -5, 4], representing an example inverse matrix A⁻¹. , so the adjoint is 3×3 matrix with rows [24, -12, -2], [5, 3, -5], [-4, 2, 4], representing an example matrix A for finding its inverse. . Since det A = 22, we get

A⁻¹ = 1/22 × matrix([24,-12,-2],[5,3,-5],[-4,2,4]) = matrix([12/11,-6/11,-1/11],[5/22,3/22,-5/22],[-2/11,1/11,2/11]) .

See also

Determinant of a matrix, cofactor