How to Find the Inverse for Any Square Matrix
In this lesson, we describe a method for finding the
inverse
of any square
matrix; and
we demonstrate the method step-by-step with examples.
Prerequisites: This material assumes familiarity with
elementary matrix operations and
echelon transformations.
How to Find the Inverse of an n x n Matrix
Let A be an n x n matrix. To find the
inverse of matrix A, we follow these steps:
- Using
elementary operators,
transform matrix A to its
reduced row echelon form, Arref.
- Inspect Arref to determine if
matrix A has an inverse.
- If Arref is equal to the
identity matrix, then matrix A is
full rank;
and matrix A has an inverse.
- If the last row of Arref is
all zeros, then matrix A is not
full rank;
and matrix A does not have an inverse.
- If A is full rank, then the inverse of matrix
A is equal to the product of the elementary operators
that produced Arref , as shown below.
A-1 =
Er Er-1 . . .
E2 E1
where
A-1 = inverse of matrix A
r = Number of elementary row operations required to
transform A to
Arref
Ei = ith elementary row operator
used to transform A to Arref
Note that the order in which elementary row operators are multiplied
is important, because
Ei Ej
is not necessarily equal to
Ej Ei.
An Example of Finding the Inverse
Let's use the above method to find the inverse of matrix
A, shown below.
The first step is to transform matrix A into its
reduced row echelon form, Arref,
using a series of
elementary row operators
Ei. We show the transformation
steps below for each elementary row operator.
- Multiply row 1 of A by -2 and add the
result to row 2 of A. This can be accomplished by pre-multiplying A by
the elementary row operator E1, which produces A1.
- Multiply row 1 of A1 by -2 and add the
result to row 3 of A1.
- Multiply row 3 of A2 by -1 and add
row 2 of A2 to
row 3 of A2.
- Add row 2 of A3 to
row 1 of A3.
- Multiply row 2 of A4 by -0.5.
- Multiply row 3 of A5 by -1
and add the result to row 2 of A5.
Note: If the operations and/or notation shown above are unclear, please review
elementary matrix operations and
echelon transformations.
The last matrix in Step 6 of the above table is
Arref, the reduced row echelon form for
matrix A. Since Arref
is equal to the identity matrix, we know that A is
full rank.
And because A is full rank, we know that A
has an inverse.
If A were less than full rank,
Arref would have all zeros in the last row; and
A would not have an
inverse.
We find the inverse of matrix A
by computing the product of the elementary operators
that produced Arref , as shown below.
A-1 =
E6 E5
E4 E3
E2 E1
In this example, we used a 3 x 3 matrix to show how to find a matrix inverse.
The same process will work on a square matrix of any size.
Test Your Understanding
Problem
Find the inverse of matrix A, shown below.
Solution
The first step is to transform matrix A into its
reduced row echelon form, Arref,
using
elementary row operators
Ei to perform
elementary row operations, as shown below.
- Multiply row 1 of A by -2 and add the
result to row 2 of A.
- Multiply row 2 of A1 by 0.5..
The last transformed matrix in the above table is
Arref,
the reduced row echelon form for
matrix A. Since the
reduced row echelon form is equal to the
identity matrix,
we know that A is
full rank.
And because A is full rank, we know that A
has an inverse.
We find the inverse by computing the product of the elementary operators
that produced Arref , as shown below.
Note: In a previous lesson, we described a
"shortcut" for
finding the inverse of a 2 x 2 matrix.