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

$\text{ The value of }\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|\text{ is }$

Let three matrices $A=\left[\begin{array}{ll}2& 1\\ 4& 1\end{array}\right]; B=\left[\begin{array}{ll}3& 4\\ 2& 3\end{array}\right]$ and $C=\left[\begin{array}{cc}3& -4\\ -2& 3\end{array}\right]$ then

$t_{\mathrm{r}}\left(\mathrm{A}\right)+t_{\mathrm{r}}\left(\frac{ABC}{2}\right)+t_{\mathrm{r}}\left(\frac{A(BC{)}^2}{4}\right)+{\mathrm{t}}_{\mathrm{r}}\left(\frac{A(BC{)}^3}{8}\right)+\dots \dots +\infty =$

Which of the following about the trace of a matrix is false?

Let $A=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{ll}{b}_1& b_2\\ b_3& b_4\end{array}\right]$. If $10A^{10}+adj\left(A^{10}\right)=B$, then $b_1+b_2+b_3+b_4$ is equal to

The matrices $P=\left[\begin{array}{lll}{u}_1& v_1& w_1\\ u_2& v_2& w_2\\ u_3& v_3& w_3\end{array}\right]$, $Q=\frac{1}{9}\left[\begin{array}{rcc}2& 2& 1\\ -12& 5& m\\ -8& 1& 5\end{array}\right]$ are such that $PQ=I$ and $\left[\begin{array}{lll}{u}_1& v_1& w_1\\ u_2& v_2& w_2\\ u_3& v_3& w_3\end{array}\right]\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}1\\ 1\\ 5\end{array}\right].$ If $y=-3,$ then the value of $m,$ is

Statement $\mathrm{I}$: Suppose $A=\left[\begin{array}{ll}1& \pi \\ 0& 1\end{array}\right],$ then $A^{100}=\left[\begin{array}{cc}1& 100\pi \\ 0& 1\end{array}\right]$
Statement II: Suppose $B$ is a $2\times 2$ matrix such that $B^2=0,$ then $(I+B{)}^n=I+nB$ for each $n\in N.$

Consider the system of linear equations

$-x+y+2z=0$

$3x-ay+5z=1$

$2x-2y-az=7$

Let $S_1$ be the set of all $a\in R$ for which the system is inconsistent and $S_2$ be the set of all $a\in R$ for which the system has infinitely many solutions. If $\mathrm{n}\left({\mathrm{S}}_1\right)$ and $\mathrm{n}\left({\mathrm{S}}_2\right)$ denote the number of elements in ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ respectively, then

If $x+ay+a^{2}z=0$,  $x+by+b^{2}z=0$, $x+cy+c^{2}z=0$ ,  $a\ne b\ne c$,  then given system of equations -