G Tewani Solutions for Chapter: Matrices, Exercise 1: Exercises

If $A$ is a $2\times 2$ matrix such that $A\left[\begin{array}{c}1\\ -1\end{array}\right]=\left[\begin{array}{c}-1\\ 2\end{array}\right]$ and $A^2\left[\begin{array}{c}1\\ -1\end{array}\right]=\left[\begin{array}{l}1\\ 0\end{array}\right]$. Then, the sum of the elements of $A$ is

If $A=\left[\begin{array}{ll}a& b\\ 0& a\end{array}\right]$ is $n^{\mathrm{th}}$ root of $I_2,$ then choose the correct statements:

$\left(\mathrm{i}\right)$ if $n$ is odd, $a=1, b=0$
$\left(\mathrm{ii}\right)$ if $n$ is odd,$a=-1, b=0$
$\left(\mathrm{iii}\right)$ if $n$ is even, $a=1, b=0$
$\left(\mathrm{iv}\right)$ if $n$ is even, $a=-1, b=0$

If $A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right]$ and $B=\left[\begin{array}{rr}1& -1\\ -1& 1\end{array}\right]$, then $A^8$ equals

If $n^{\mathrm{th}}$ order square matrix $A$ is an orthogonal, then $\left|\mathrm{adj}\left(adjA\right)\right|$ is

If $A={\left[a_{ij}\right]}_{4\times 4},$ such that $a_{ij}=\left\{\begin{array}{ll}2,& \text{when} i=j\\ 0,& \text{when} i\ne j\end{array}\right.$, then $\left\{\frac{det\left(adj\left(adjA\right)\right)}{7}\right\}$ is (where $\left\{.\right\}$ represents fractional part function)

Let $f\left(x\right)=\frac{1+x}{1-x}$. If $A$ is matrix for which $A^3=O,$ then $f\left(A\right)$ is

If $A^2-A+I=0,$ then the inverse of $A$ is

If $AB=A$ and $BA=B$, then which of the following is/are true?