Matrix Inverse Solved Examples
Let’s solve some example questions on Inverse of Matrix.
Example 1: Find the inverse of the matrix [Tex]\bold{A=\left[\begin{array}{ccc}2 & 3 & 1\\1 & 1 & 2\\2 & 3 & 4\end{array}\right]}[/Tex] using the formula.
Solution:
We have,
[Tex]A=\left[\begin{array}{ccc}2 & 3 & 1\\1 & 1 & 2\\2 & 3 & 4\end{array}\right] [/Tex]
Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.
adj A = [Tex]\left[\begin{array}{ccc}-2 & -9 & 5\\0 & 6 & -3\\1 & 0 & -1\end{array}\right] [/Tex]
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 = [Tex]\frac{1}{-3}\left[\begin{array}{ccc}-2 & -9 & 5\\0 & 6 & -3\\1 & 0 & -1\end{array}\right] [/Tex]
= [Tex]\left[\begin{array}{ccc}\frac{2}{3} & 3 & – \frac{5}{3}\\0 & -2 & 1\\- \frac{1}{3} & 0 & \frac{1}{3}\end{array}\right] [/Tex]
Example 2: Find the inverse of the matrix A=\bold{ using the formula.}[Tex]\left[\begin{array}{ccc}6 & 2 & 3\\0 & 0 & 4\\2 & 0 & 0\end{array}\right] [/Tex]
Solution:
We have,
A=[Tex]\left[\begin{array}{ccc}6 & 2 & 3\\0 & 0 & 4\\2 & 0 & 0\end{array}\right] [/Tex]
Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.
adj A = [Tex]\left[\begin{array}{ccc}0 & 0 & 8\\8 & -6 & -24\\0 & 4 & 0\end{array}\right] [/Tex]
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 = [Tex]\frac{1}{16}\left[\begin{array}{ccc}0 & 0 & 8\\8 & -6 & -24\\0 & 4 & 0\end{array}\right] [/Tex]
= [Tex]\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] [/Tex]
Example 3: Find the inverse of the matrix A=[Tex]\bold{\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & 4\\0 & 0 & 1\end{array}\right] } [/Tex] using the formula.
Solution:
We have,
A=[Tex]\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & 4\\0 & 0 & 1\end{array}\right] [/Tex]
Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.
adj A = [Tex]\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex]
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 = [Tex]\frac{1}{1}\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex]
= [Tex]\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex]
Example 4: Find the inverse of the matrix A=[Tex]\bold{\left[\begin{array}{ccc}1 & 2 & 3\\2 & 1 & 4\\3 & 4 & 1\end{array}\right] } [/Tex] using the formula.
Solution:
We have,
A=[Tex]\left[\begin{array}{ccc}1 & 2 & 3\\2 & 1 & 4\\3 & 4 & 1\end{array}\right] [/Tex]
Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.
adj A = [Tex]\left[\begin{array}{ccc}-15 & 10 & 5\\10 & -8 & 2\\5 & 2 & -3\end{array}\right] [/Tex]
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 = [Tex]\frac{1}{20}\left[\begin{array}{ccc}-15 & 10 & 5\\10 & -8 & 2\\5 & 2 & -3\end{array}\right] [/Tex]
= [Tex]\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] [/Tex]
Inverse of a Matrix
The inverse of Matrix is the matrix that on multiplying with the original matrix results in an identity matrix. For any matrix A, its inverse is denoted as A-1.
Let’s learn about the Matrix Inverse in detail, including its definition, formula, methods on how to find the inverse of a matrix, and examples.
Table of Content
- Matrix Inverse
- Terms Related to Matrix Inverse
- How to Find Inverse of Matrix?
- Inverse of a Matrix Formula
- Inverse Matrix Method
- Inverse of 2×2 Matrix Example
- Determinant of Inverse Matrix
- Properties of Inverse of Matrix
- Matrix Inverse Solved Examples