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.

Test Your Understanding

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

The correct answer is (D).

  • The rank of a matrix is defined as the maximum number of linearly independent row vectors in the matrix. Rows 1 and 2 of matrix A are not independent, since Row 1 = 2 * Row 2. Therefore, A has only one independent row, so its rank is 1. Previously, we described how to compute matrix rank.
  • Previously, we showed how to find the determinant of a 2 x 2 matrix. We use that approach to find the determinant of A, which is denoted |A|.

    |A| = ( A11 * A22 ) - ( A12 * A21 )
    |A| = ( 2 * 2 ) - ( 4 * 1 ) = 4 - 4 = 0

  • Matrix A is not a full rank matrix. And its determinant is equal to zero. Therefore, matrix A does not have an inverse, which means that matrix A is singular.

Note: If a square matrix is less than full rank, its determinant is equal to zero; and vice versa.