Matrix

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. Nonsquare 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 since

, its inverse is

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:

2. Augmented matrix method Use Gauss-Jordan elimination to transform [ A | I ] into [ I | A-1 ].

Example: The following steps result in

.

so we see that

.

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

.

The cofactor matrix for A is

, so the adjoint

is

. Since det A = 22, we get

Determinant A single number obtained from a matrix that reveals a variety of the matrix's properties. Determinants of small matrices are written and evaluated as shown below. Determinants may also be found using expansion by cofactors. Note: Although a determinant looks like an absolute value it is not. The determinant of a matrix may be negative or positive.

Cofactor Matrix Matrix of Cofactors A matrix with elements that are the cofactors, term-by-term, of a given square matrix.