# Matrix Inverse

This lesson defines the matrix inverse, and shows how to determine whether the inverse of a matrix exists.

## Matrix Inversion

Suppose A is an n x n matrix. The inverse of A is another n x n matrix, denoted A-1, that satisfies the following conditions.

AA-1 = A-1A = In

where In is the identity matrix. Below, with an example, we illustrate the relationship between a matrix and its inverse.

 2 1 3 4
 0.8 -0.2 -0.6 0.4
=
 1 0 0 1
A A-1 I

 0.8 -0.2 -0.6 0.4
 2 1 3 4
=
 1 0 0 1
A-1 A I

Not every square matrix has an inverse; but if a matrix does have an inverse, it is unique.

## Does the Inverse Exist?

There are two ways to determine whether the inverse of a square matrix exists.

• Determine its rank. The rank of a matrix is a unique number associated with a square matrix. If the rank of an n x n matrix is less than n, the matrix does not have an inverse. We showed how to determine matrix rank previously.
• Compute its determinant. The determinant is another unique number associated with a square matrix. When the determinant for a square matrix is equal to zero, the inverse for that matrix does not exist. We showed how to find the determinant of a matrix previously.

A square matrix that has an inverse is said to be nonsingular or invertible; a square matrix that does not have an inverse is said to be singular.

Problem 1

Consider the matrix A, shown below.

A =
 2 4 1 2

Which of the following statements are true?

(A) The rank of matrix A is 1.
(B) The determinant of matrix A is 0.
(C) Matrix A is singular.
(D) All of the above.
(E) None of the above.

Solution