Inverse of a Matrix (Practice Problems)

The inverse of a matrix is a matrix that, when multiplied by the original matrix, produces the identity matrix. It is similar to the reciprocal of a number.

Solved Examples

Example 1: Find the inverse of the matrix \bold{A=\left[\begin{array}{ccc}2 & 3 & 1\\1 & 1 & 2\\2 & 3 & 4\end{array}\right]}using the formula.

Solution:

We have,

A=\left[\begin{array}{ccc}2 & 3 & 1\\1 & 1 & 2\\2 & 3 & 4\end{array}\right]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix's transpose.

adj A = \left[\begin{array}{ccc}-2 & -9 & 5\\0 & 6 & -3\\1 & 0 & -1\end{array}\right]

Find the value of determinant of the matrix.

|A| = 2(4–6) – 3(4–4) + 1(3–2)

= –3

So, the inverse of the matrix is,

A^–1 = \frac{1}{-3}\left[\begin{array}{ccc}-2 & -9 & 5\\0 & 6 & -3\\1 & 0 & -1\end{array}\right]

      = \left[\begin{array}{ccc}\frac{2}{3} & 3 & - \frac{5}{3}\\0 & -2 & 1\\- \frac{1}{3} & 0 & \frac{1}{3}\end{array}\right]

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

Solution:

We have,

A=\left[\begin{array}{ccc}6 & 2 & 3\\0 & 0 & 4\\2 & 0 & 0\end{array}\right]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix's transpose.

adj A = \left[\begin{array}{ccc}0 & 0 & 8\\8 & -6 & -24\\0 & 4 & 0\end{array}\right]

Find the value of determinant of the matrix.

|A| = 6(0–4) – 2(0–8) + 3(0–0)

     = 16

So, the inverse of the matrix is,

A^–1 = \frac{1}{16}\left[\begin{array}{ccc}0 & 0 & 8\\8 & -6 & -24\\0 & 4 & 0\end{array}\right]

      = \left[\begin{array}{ccc}0 & 0 & \frac{1}{2}\\\frac{1}{2} & - \frac{3}{8} & - \frac{3}{2}\\0 & \frac{1}{4} & 0\end{array}\right]

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

Solution:

We have,

A=\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & 4\\0 & 0 & 1\end{array}\right]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix's transpose.

adj A = \left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right]

Find the value of determinant of the matrix.

|A| = 1(1–0) – 2(0–0) + 3(0–0)

= 1

So, the inverse of the matrix is,

A^–1 = \frac{1}{1}\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right]

= \left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right]

Example 4: Find the inverse of the matrix A=\bold{\left[\begin{array}{ccc}1 & 2 & 3\\2 & 1 & 4\\3 & 4 & 1\end{array}\right]} using the formula.

Solution:

We have,

A=\left[\begin{array}{ccc}1 & 2 & 3\\2 & 1 & 4\\3 & 4 & 1\end{array}\right]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix's transpose.

adj A = \left[\begin{array}{ccc}-15 & 10 & 5\\10 & -8 & 2\\5 & 2 & -3\end{array}\right]

Find the value of determinant of the matrix.

|A| = 1(1–16) – 2(2–12) + 3(8–3)

= 20

So, the inverse of the matrix is,

A^–1 = \frac{1}{20}\left[\begin{array}{ccc}-15 & 10 & 5\\10 & -8 & 2\\5 & 2 & -3\end{array}\right]

      = \left[\begin{array}{ccc}- \frac{3}{4} & \frac{1}{2} & \frac{1}{4}\\\frac{1}{2} & - \frac{2}{5} & \frac{1}{10}\\\frac{1}{4} & \frac{1}{10} & - \frac{3}{20}\end{array}\right]

Practice Problem

Question 1. Find the inverse of the matrix A = \begin{bmatrix} 3 & 2 \\ 4 & 1 \end{bmatrix}.

Question 2. Determine if the following matrix has an inverse. If yes, find the inverse A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}.

Question 3. Find the inverse of the matrix A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}.

Question 4. Verify if the following matrix has an inverse. If the inverse exists, compute it A = \begin{bmatrix} 2 & 3 \\ 4 & 6 \end{bmatrix}.

Answer:-

1. A^{-1} = \begin{bmatrix} -\frac{1}{5} & \frac{2}{5} \\ \frac{4}{5} & -\frac{3}{5} \end{bmatrix}

2. A^{-1} = \begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}

3. A^{-1} = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix}

4. does not have an inverse since the determinant is 0.