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