Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 3: Exercise-3

Let $M$ be a $3\times 3$ matrix satisfying $M\left[\begin{array}{l}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right], M\left[\begin{array}{c}1\\ -1\\ 0\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ -1\end{array}\right]$ and $M \left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 12\end{array}\right]$. Then the sum of the diagonal entries of $M$ is

Let $X$ and $Y$ be two arbitrary, $3\times 3,$ non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3\times 3,$ non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Which of the following values of $\mathrm{\alpha }$ satisfy the equation $\left|\begin{array}{ccc}(1+\mathrm{\alpha }{)}^2& (1+2\mathrm{\alpha }{)}^2& (1+3\mathrm{\alpha }{)}^2\\ (2+\mathrm{\alpha }{)}^2& (2+2\mathrm{\alpha }{)}^2& (2+3\mathrm{\alpha }{)}^2\\ (3+\mathrm{\alpha }{)}^2& (3+2\mathrm{\alpha }{)}^2& (3+3\mathrm{\alpha }{)}^2\end{array}\right|=-648\mathrm{\alpha }$?

How many $3\times 3$ matrices $M$ with entries from $\left\{0, 1, 2\right\}$ are there, for which the sum of the diagonal entries of $M^{T}M$ is $5?$

Let $A$ be a $2\times 2$ matrix

Statement $1:$ $adj\left(adjA\right)=A$

Statement $2:$ $\left|adjA\right|=|A\mid$.

If $A$ is an $3\times 3$ non-singular matrix such that $A{A}^{\text{'}}=A^{\text{'}}A$ and $B=A^{-1}{A}^{\text{'}},$ then $B{B}^{\text{'}}$ equals

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$ is a matrix satisfying the equation $A{A}^T=9I,$ where $I$ is $3\times 3$ identity matrix, then the ordered pair $\left(a,b\right)$ is equal to 

If $A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right]$, then $\mathrm{adj}\left(3A^2+12A\right)$ is equal to