R D Sharma Solutions for Chapter: Adjoint and Inverse of a Matrix, Exercise 1: EXERCISE

Find the adjoint of the matrix $A=\left[\begin{array}{cc}-3& 5\\ 2& 4\end{array}\right]$

Verify that $\therefore (adjA)A=\left|A\right|I=A\left(\mathrm{adj}A\right)$

Find the adjoint of the following matrix:

$B=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$

Verify that $: (adjB)B=\left|B\right|I=B\left(\mathrm{adj}B\right)$

If $A=\left[\begin{array}{ll}2& 3\\ 1& 2\end{array}\right]$ verify that $A^2-4A+I=O,\text{ where }I=\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right]$ and $O=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$. Hence find $A^{-1}$

Show that $\mathit{A}=\left[\begin{array}{cc}-8& 5\\ 2& 4\end{array}\right]$ satisfy the equation $A^2+4A-42I=0$ Hence find $A^{-1}$

Find the matrix $X$ for which

$\left[\begin{array}{ll}3& 2\\ 7& 5\end{array}\right]X\left[\begin{array}{ll}-1& 1\\ -2& 1\end{array}\right]=\left[\begin{array}{cc}2& -1\\ 0& 4\end{array}\right]$

If $A$ is a square matrix of order $n$, prove that $| A \mathrm{adj} A|={\left|A\right|}^n$.

If $A=\left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 4\\ -2& 2& 1\end{array}\right],\text{ find }{\left(A^T\right)}^{-1}$

If $A=\left[\begin{array}{lll}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]$ find $A^{-1}$ and show that  $A^{-1}=\frac{1}{2}\left(A^2-3I\right)$