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^{-1}A =
I_{n}
where I_{n} is the
identity matrix.
Below, with an example, we illustrate the relationship between a matrix
and its inverse.
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 of This Lesson
Problem 1
Consider the matrix A, shown below.
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|
= ( A_{1}_{1} * A_{2}_{2} )
- ( A_{1}_{2} * A_{2}_{1} )
|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.