Find Matrix Inverse Using Row Operations

Introduction

We present examples on how to find the inverse of a matrix using the three row operations listed below:

1. Interchange two rows
2. Add a multiple of one row to another
3. Multiply a row by a non zero constant

Inverse of a Matrix

Let A be an n × n matrix. If matrix A^-1 is the inverse of matrix A , then we have

A A^-1 = I_n = A^-1 A

[ I_n | A^-1 ]

Examples with Solutions

Example 1
Find the inverse of matrix

Solution to Example 1
Write the augmented matrix [ A | I₂ ]

Let R₁ and R₂ be the first and the second rows of the above augmented matrix.
Write the above augmented matrix in reduced row echelon form .

The above augmented matrix has the form [ I₂ | A^-1 ] and therefore A^-1 is given by

Example 2
Find the inverse of matrix

Solution to Example 2
Write the augmented matrix [ A | I₃ ]


Let R₁, R₂ and R₃ be the first, the second and the third rows respectively of the above augmented matrix.
Write the above augmented matrix in reduced row echelon form .
$$$$$$

The above augmented matrix has the form $[I_3|A^{-1}]$ and therefore $A^{-1}$ is given by
$A^{-1}=\left[\begin{array}{ccc}-\frac{2}{3}& \frac{2}{3}& -\frac{1}{3}\\ 2& -1& 1\\ -\frac{1}{3}& \frac{1}{3}& \frac{1}{3}\end{array}\right]$

Example 3
Find the inverse of matrix $A=\left[\begin{array}{cccc}0& 1& -1& 1\\ 2& 2& 0& -2\\ 1& 1& -2& 0\\ 0& 1& 2& 0\end{array}\right]$.

Solution to Example 3
Write the augmented matrix $[A|I_4]$
$\left[\begin{array}{ccccccccc}0& 1& -1& 1& |& 1& 0& 0& 0\\ 2& 2& 0& -2& |& 0& 1& 0& 0\\ 1& 1& -2& 0& |& 0& 0& 1& 0\\ 0& 1& 2& 0& |& 0& 0& 0& 1\end{array}\right]$
Write the above augmented matrix in reduced row echelon form .
Interchange $R_1$ and $R_3$


Interchange $R_2$ and $R_4$

The above augmented matrix has the form $[I_4|A^{-1}]$ and therefore $A^{-1}$ is given by
$A^{-1}=\left[\begin{array}{cccc}-4& -2& 5& 3\\ 2& 1& -2& -1\\ -1& -1/2& 1& 1\\ -2& -3/2& 3& 2\end{array}\right]$

Example 4
Find the inverse of matrix $A=\left[\begin{array}{ccc}1& 2& 0\\ 0& 1& 3\\ 1& 4& 6\end{array}\right]$.

Solution to Example 4
Write the augmented matrix $[A|I_3]$
$\left[\begin{array}{ccccccc}1& 2& 0& |& 1& 0& 0\\ 0& 1& 3& |& 0& 1& 0\\ 1& 4& 6& |& 0& 0& 1\end{array}\right]$
Write the above augmented matrix in reduced row echelon form .
$\begin{array}{c}\\ \\ R_3-R_1\end{array}\left[\begin{array}{ccccccc}1& 2& 0& |& 1& 0& 0\\ 0& 1& 3& |& 0& 1& 0\\ 0& 2& 6& |& -1& 0& 1\end{array}\right]$

$\begin{array}{c}\\ \\ R_3-2R_2\end{array}\left[\begin{array}{ccccccc}1& 2& 0& |& 1& 0& 0\\ 0& 1& 3& |& 0& 1& 0\\ 0& 0& 0& |& -1& -2& 1\end{array}\right]$

The last row of the original matrix (on the left side) is all zeros and therefore the rows in the orgiginal matrix $A$ are not linearly independent and hence the given matrix is NOT invertible.

More References and links

1. matrix inverse
2. linear algebra
3. identity matrix
4. Find the Inverse of a Matrix Using Row Reduction - Calculator - Calculator
5. Solve a system of linear equations by elimination
6. elementary matrices