Inverse Matrix Questions with Solutions

Tutorials including examples and questions with detailed solutions on how to find the inverse of square matrices using the method of the row echelon form and the method of cofactors. The properties of inverse matrices are discussed and various questions, including some challenging ones, related to inverse matrices are included along with their detailed solutions.

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Definition of the Identity Matrix

The identity matrix I n is the square matrix with order n x n and with the elements in the main diagonal consisting of 1's and all other elements are equal to zero.
Examples of indentity matrices
identity matrices


Definition of The Inverse of a Matrix

Let A be a square matrix of order n x n. If there exists a matrix B of the same order such that
A B = I n = B A
where I n is the identity matrix of order n x n, then B is called the inverse matrix of A and matrix A is the inverse matrix of B.
Example 1
Verify that matrices A and B given below are inverses of each other.
2 by 2 Matrix for Example 1

Solution
Let us find the products AB and BA
Product of 2 by 2 Matrice for Example 1

AB = BA = I 2 and therefore A and B are inverse of each other.
The inverse of a square matrix A is denoted as A -1 and is unique.


Find the Inverse of a Square Matrix Using the Row Reduction Method

This method is also called Gauss–Jordan elimination method.
We first write the augmented matrix \( \) \( \) \( \) \( \) \[ [ A | I ] \] where I is the identity matrix of the same order as matrix A and then use row elementary operations to rewrite it as \[ [ I | A^{-1} ] \] where \( A^{-1} \) is the inverse of matrix A.
An online calculator to find the inverse of a matrix using row reduction is included.

Example 2
Find the inverse of matrix A given by \[ A = \begin{bmatrix} 1&1 \\ 2&4 \end{bmatrix} \] if it exists.
Solution
Write the augmented matrix \( [ A | I )\) \[ \begin{bmatrix} 1&1&|&1&0\\2&4&|&0&1 \end{bmatrix} \] step 1 \[ \color{red}{\begin{matrix} \\ R_2 - 2 \times R_1 \end{matrix} } \begin{bmatrix} 1&1&|&1&0\\0&2&|&-2&1 \end{bmatrix} \] step 2 \[ \color{red}{\begin{matrix} \\ (1/2)R_2 \end{matrix} } \begin{bmatrix} 1&1&|&1&0\\0&1&|&-1&1/2 \end{bmatrix} \] step 3 \[ \color{red}{\begin{matrix} R_1 - R_2 \\ \\ \end{matrix} } \begin{bmatrix} 1&0&|&2&-1/2\\0&1&|&-1&1/2 \end{bmatrix} \] The inverse of A is the 2 × 2 matrix on the right side given by \[ A^{-1} = \begin{bmatrix} 2&-1/2\\-1&1/2 \end{bmatrix} \]

Example 3
Find the inverse of matrix A given by
\[ A = \begin{bmatrix}-2&2&0 \\ 2&1&3\\ -2&4&-2\end{bmatrix} \] if it exists.
Solution
Write the augmented matrix \( [ A | I )\) \[ \begin{bmatrix} -2&2&0&|&1&0&0\\ 2&1&3&|&0&1&0 \\ -2 & 4 & -2 &|& 0 & 0 & 1 \end{bmatrix} \] step 1 \[ \color{red}{ \begin{matrix} \\ R_2 + R_1 \\ R_3 - R_1 \end{matrix} } \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&3&3&|&1&1&0 \\ 0 & 2 & -2 &|& -1 & 0 & 1 \end{bmatrix} \] step 2 \[ \color{red}{ \begin{matrix} \\ \\ R_3 - (2/3) R_2 \\ \end{matrix} } \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&3&3&|&1&1&0 \\ 0 & 0 & - 4&|& -5/3 & -2/3 & 1 \end{bmatrix} \] step 3 \[ \color{red}{ \begin{matrix} \\ \\ (-1/4)R_3 \\ \end{matrix}} \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&3&3&|&1&1&0 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] step 4 \[ \color{red}{ \begin{matrix} \\ R_2 - 3\times R_3 \\ \\ \end{matrix} } \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&3&0&|&-1/4&1/2&3/4 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] step 5 \[ \color{red}{ \begin{matrix} \\ (1/3) R_2 \\ \\ \end{matrix}} \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&1&0&|&-1/12&1/6&1/4 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] step 6 \[ \color{red}{ \begin{matrix} R_1- 2\times R_2 \\ \\ \\ \end{matrix} } \begin{bmatrix} -2&0&0&|&7/6&-1/3&-1/2\\ 0&1&0&|&-1/12&1/6&1/4 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] step 7 \[ \color{red}{ \begin{matrix} (-1/2) R_1 \\ \\ \\ \end{matrix} } \begin{bmatrix} 1&0&0&|&-7/12&1/6&1/4\\ 0&1&0&|&-1/12&1/6&1/4 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] Hence \[ A^{-1} = \begin{bmatrix} -7/12&1/6&1/4\\ -1/12&1/6&1/4 \\ 5/12 & 1/6 & -1/4 \end{bmatrix} \]
More examples on how to find matrix inverse using row operations are included.


Find the Inverse of a Square Matrix Using Minors, Cofactors and Adjugate

This method is explained using a numerical example. Matrix A is given below.
\[ A = \begin{bmatrix} -1&0&1\\ 2&-1&2 \\ -1 & 2 & 1 \end{bmatrix} \] a) Find the matrices of minors and cofactors, the adjugate and the inverse of A.
Matrix of Minors
The entry \( M_{i,j} \) of the matrix of minors of matrix A is given by the determinant obtained by deleting the \( i^{th}\) row and the \( j^{th}\) column.

To find \( M_{1,1} \), delete row 1 and column 1 from matrix A and find the determinant of the remaining 2 by 2 matrix as follows: \( M_{1,1} = Det \begin{bmatrix} .&.&.\\ .&-1&2 \\ .& 2 & 1 \end{bmatrix} = -1 - 4 = -5\)

To find \( M_{1,2} \), delete row 1 and column 2 from matrix A and find the determinant of the remaining 2 by 2 matrix as follows: \( M_{1,2} = Det \begin{bmatrix} .&.&.\\ 2&.&2 \\ -1 & . & 1 \end{bmatrix} = 2 -(-2) = 4 \)

To find \( M_{1,3} \), delete row 1 and column 3 from matrix A and find the determinant of the remaining 2 by 2 matrix as follows: \( M_{1,3} = Det \begin{bmatrix} .&.&.\\ 2&-1&. \\ -1 & 2 & . \end{bmatrix} = 4 - 1 = 3 \)

To find \( M_{2,1} \), delete row 2 and column 1 from matrix A and find the determinant of the remaining 2 by 2 matrix as follows: \( M_{2,1} = Det \begin{bmatrix} .&0&1\\ .&.&. \\ . & 2 & 1 \end{bmatrix} = 0 - 2 = - 2 \)
...
...
The remaining entries are given by: \( M_{2,2} = 0 \) , \( M_{2,3} = -2 \) , \( M_{3,1} = 1\) , \( M_{3,2} = -4\) , \( M_{3,3} = 1\).
The matrix of minors M is given by
\( M = \begin{bmatrix} -5&4&3\\ -2&0&-2\\ 1&-4&1 \end{bmatrix} \)
Matrix of Cofactors
The entries \( C_{i,j} \) of the matrix of cofactors C of matrix A are given by \( C_{i,j} = (-1)^{i+j}M{i,j} \)

An evaluation of the entries \( C_{i,j} \) gives:
\( C_{1,1} = (-1)^{1+1} M_{1,1} = -5 \)
\( C_{1,2} = (-1)^{1+2} M_{1,2} = - 4 \)
\( C_{1,3} = (-1)^{1+3} M_{1,3} = 3 \)
\( C_{2,1} = (-1)^{2+1} M_{2,1} = 2 \)
\( C_{2,2} = (-1)^{2+2} M_{2,2} = 0 \)
\( C_{3,1} = (-1)^{3+1} M_{3,1} = 1 \)
\( C_{3,2} = (-1)^{3+2} M_{3,2} = 4 \)
\( C_{3,3} = (-1)^{3+3} M_{3,1} = 1 \)
Hence the matrix C of cofactors is given by \( C = \begin{bmatrix} -5&-4&3\\ 2&0&2\\ 1&4&1 \end{bmatrix} \)
Adjugate (or adjunct) of a Matrix
The adjugate (or adjunct) of matrix A is the transpose of its matrix of cofactors C.
\( Adjugate(A) = C^T = \begin{bmatrix} -5&2&1\\ -4&0&4\\ 3&2&1 \end{bmatrix} \)
Inverse Matrix
We now need to find the determinant D of matrix A.
Using the first row of matrix A and the corresponding minors already found, D is given by
\( D = det\begin{bmatrix} -1&0&1\\ 2&-1&2 \\ -1 & 2 & 1 \end{bmatrix} = A_{11}M_{1,1} - A_{1,2}M_{1,2} + A_{1,3}M_{1,3} = 8\)
The inverse of \( A \) is given by \( A^{-1} = \dfrac{1}{D} C^T = \dfrac{1}{8} \begin{bmatrix} -5&2&1\\ -4&0&4\\ 3&2&1 \end{bmatrix} = \begin{bmatrix} -\dfrac{5}{8}&\dfrac{1}{4}&\dfrac{1}{8}\\ -\dfrac{1}{2}&0&\dfrac{1}{2}\\ \dfrac{3}{8}&\dfrac{1}{4}&\dfrac{1}{8}\end{bmatrix}\)


Formula for the Inverse of a 2 by 2 Matrix

Using any of the two methods described above, it can be shown that the inverse of matrix A given by \[ A = \begin{bmatrix} a & b\\ c & d \\ \end{bmatrix} \] is given by \[ A^{-1} = \dfrac{1}{ad - bc}\begin{bmatrix} d & -b\\ -c & a \\ \end{bmatrix} \]


Properties of Inverse Matrices

A matrix that has an inverse is called an invertible matrix.

  1. If A is an invertible matrix, its inverse is unique.
  2. \( A A^{-1} = A^{-1} A = I \)
  3. If matrices A and B are invertible, then:\( (AB)^{-1} = B^{-1}A^{-1} \)
  4. A matrix is invertible if and only if its determinant is not equal to zero.
  5. A matrix whose determinant is not equal to zero is called nonsingular.
  6. \( (A^T)^{-1} = (A^{-1})^T \)
  7. \( Det(A^{-1}) = \dfrac{1}{Det(A)} \)
  8. \( (A^{-1})^{-1} = A \)


Questions on Inverse Matrices



Solutions to the Above Questions


More References and Links to Matrices