Question

Find the inverse of the matrix $A=\left[ \begin{matrix}
   1 & 2 \\
   2 & 1 \\
\end{matrix} \right]$ .

Answer 1

Hint: We have a formula from which we can find the inverse of any matrix provided, the determinant of that matrix is not equal to $0$. The formula to find the inverse of the matrix $A$ is ${{A}^{-1}}=\dfrac{1}{\left| A \right|}adj\left( A \right)$ where $\left| A \right|$ is the determinant and $adj\left( A \right)$ is the adjoint of matrix $A$.

Complete step-by-step solution -
Before proceeding with the question, we must know all the formulas that will be required to solve this question.
The inverse of a matrix $A$ is given by the formula,
${{A}^{-1}}=\dfrac{1}{\left| A \right|}adj\left( A \right)..............\left( 1 \right)$
Here $\left| A \right|$ is the determinant and $adj\left( A \right)$ is the adjoint of matrix $A$.
The determinant of a matrix $A=\left[ \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right]$ is given by the formula,
$\left| A \right|=ad-bc.................\left( 2 \right)$
To find the adjoint of the matrix, we will first find the cofactor matrix which is given by,
$\left[ \begin{matrix}
   d & -c \\
   -b & a \\
\end{matrix} \right].......................\left( 3 \right)$
The adjoint of the matrix can be found by taking the transpose of the cofactor matrix. So, the transpose of the matrix is given by,
$adj\left( A \right)=\left[ \begin{matrix}
   d & -b \\
   -c & a \\
\end{matrix} \right]$
In the question, we are given a matrix $A=\left[ \begin{matrix}
   1 & 2 \\
   2 & 1 \\
\end{matrix} \right]$. We have to find the inverse of this matrix.
Using formula $\left( 2 \right)$, the determinant of this matrix is,
$\left| A \right|=1-4$
$\Rightarrow \left| A \right|=-3..............(4)$
The adjoint of the matrix can be found by using the steps shown in the above paragraph,
We have a matrix $A=\left[ \begin{matrix}
   1 & 2 \\
   2 & 1 \\
\end{matrix} \right]$. Using formula (3), the cofactor matrix for this matrix will be,
$\left[ \begin{matrix}
   1 & -2 \\
   -2 & 1 \\
\end{matrix} \right]$
The adjoint of the matrix can be found by taking the transpose of the matrix and will be equal to,
$adj\left( A \right)=\left[ \begin{matrix}
   1 & -2 \\
   -2 & 1 \\
\end{matrix} \right]$
Since we have got the determinant and the adjoint of the matrix, we can now find it’s inverse. Using formula $\left( 1 \right)$, we get,
${{A}^{-1}}=\dfrac{1}{-3}\left[ \begin{matrix}
   1 & -2 \\
   -2 & 1 \\
\end{matrix} \right]$
Now, we know if we take $\dfrac{-1}{3}$ inside the matrix, it gets multiplied with each element of the matrix.
$\therefore {{A}^{-1}}=\left[ \begin{matrix}
   \dfrac{-1}{3} & \dfrac{2}{3} \\
   \dfrac{2}{3} & \dfrac{-1}{3} \\
\end{matrix} \right]$

Note: There is a possibility that one may commit a mistake while finding the adjoint of the matrix. It is a very common mistake that one does not take the transpose of the cofactor matrix while finding the adjoint of the matrix, and this leads us to an incorrect answer.