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 stepbystep 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, A_{rref}.
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_{r1} . . .
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.
The first step is to transform matrix A into its
reduced row echelon form, A_{rref},
using a series of
elementary row operators
E_{i}. We show the transformation
steps below.
Elementary row operation 
Row operator, E_{i} 
Transformed matrix, A_{i} 
1. Multiply row 1 of A by 2 and add the
result to row 2 of A



2. Multiply row 1 of A_{1} by 2 and add the
result to row 3 of A_{1}



3. Multiply row 3 of A_{2} by 1 and add
row 2 of A_{2} to
row 3 of A_{2}



4. Add row 2 of A_{3} to
row 1 of A_{3}



5. Multiply row 2 of A_{4} by 0.5



6. Multiply row 3 of A_{5} by 1
and add the result to row 2 of A_{5}



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
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} = 







E_{6} 
E_{5} 
E_{4} 
E_{3} 
E_{2} 
E_{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.
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, A_{rref},
using
elementary row operators
E_{i} to perform
elementary row operations, as shown below.
Elementary row operation 
Row operator, E_{i} 
Transformed matrix, A_{i} 
1. Multiply row 1 of A by 2 and add the
result to row 2 of A



2. Multiply row 2 of A_{1} by 0.5.



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.
A^{1} =
E_{2} E_{1} =



= 


E_{2} 
E_{1} 


Note: In a previous lesson, we described a
"shortcut" for
finding the inverse of a 2 x 2 matrix.