Inspect A_{rref} to determine if
matrix A has an inverse.
If A_{rref} is equal to the
identity matrix, then matrix A is
full rank;
and matrix A has an inverse.
If the last row of A_{rref} 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 A_{rref} , as shown below.
A^{-1} =
E_{r}E_{r-1} . . .
E_{2}E_{1}
where
A^{-1} = inverse of matrix A r = Number of elementary row operations required to
transform A to
A_{rref} E_{i} = ith elementary row operator
used to transform A to A_{rref}
Note that the order in which elementary row operators are multiplied
is important, because
E_{i}E_{j}
is not necessarily equal to
E_{j}E_{i}.
An Example of Finding the Inverse
Let's use the above method to find the inverse of matrix
A, shown below.
A =
1
2
2
2
2
2
2
2
1
The first step is to transform matrix A into its
reduced row echelon form, A_{rref},
using a series of
elementary row operatorsE_{i}. 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 E_{1}, which produces A_{1}.
E_{1} =
1
0
0
-2
1
0
0
0
1
A_{1} = E_{1}A =
1
2
2
0
-2
-2
2
2
1
Multiply row 1 of A_{1} by -2 and add the
result to row 3 of A_{1}.
E_{2} =
1
0
0
0
1
0
-2
0
1
A_{2} = E_{2}A_{1} =
1
2
2
0
-2
-2
0
-2
-3
Multiply row 3 of A_{2} by -1 and add
row 2 of A_{2} to
row 3 of A_{2}.
E_{3} =
1
0
0
0
1
0
0
1
-1
A_{3} = E_{3}A_{2} =
1
2
2
0
-2
-2
0
0
1
Add row 2 of A_{3} to
row 1 of A_{3}.
E_{4} =
1
1
0
0
1
0
0
0
1
A_{4} = E_{4}A_{3} =
1
0
0
0
-2
-2
0
0
1
Multiply row 2 of A_{4} by -0.5.
E_{5} =
1
0
0
0
-0.5
0
0
0
1
A_{5} = E_{5}A_{4} =
1
0
0
0
1
1
0
0
1
Multiply row 3 of A_{5} by -1
and add the result to row 2 of A_{5}.
The last matrix in Step 6 of the above table is
A_{rref}, the reduced row echelon form for
matrix A. Since A_{rref}
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,
A_{rref} 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 A_{rref} , as shown below.
A^{-1} =
E_{6}E_{5}E_{4}E_{3}E_{2}E_{1}
A^{-1} =
-1
1
0
1
-1.5
1
0
1
-1
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.
Multiply row 1 of A by -2 and add the
result to row 2 of A.
E_{1} =
1
0
-2
1
A_{1} = E_{1}A =
1
0
0
2
Multiply row 2 of A_{1} by 0.5..
E_{2} =
1
0
0
0.5
A_{rref} = E_{2}A_{1} =
1
0
0
1
The last transformed matrix in the above table is
A_{rref},
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 A_{rref} , as shown below.