Embibe Experts Solutions for Chapter: Determinants and Matrices, Exercise 1: Exercise 1

If  $a, b, c$ are $pth$, $qth$ and $rth$ terms of a GP, then $\left|\begin{array}{lll}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|$ is equal to

$\left|\begin{array}{ccc}{b}^2+c^2& a^2& a^2\\ b^2& c^2+a^2& b^2\\ c^2& c^2& a^2+b^2\end{array}\right|=$

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

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

If $\left|\begin{array}{ccc}1+ax& 1+bx& 1+cx\\ 1+a_{1}x& 1+b_{1}x& 1+c_{1}x\\ 1+a_{2}x& 1+b_{2}x& 1+c_{2}x\end{array}\right|=A_0+A_{1}x+A_2{x}^2+A_3{x}^3$ , then $A_0$ is

The greatest value of $x$ satisfying the equation $\left|\begin{array}{ccc}{2}^{11}-x& -2^{12}& 2^{11}\\ -2^{12}& 2^{11}-x& 2^{11}\\ 2^{11}& 2^{11}& -2^{12}-x\end{array}\right|=0$ is

If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$, then$A^2+B^2$ is equal to

Let $A^{-1}=BC$ then the value of $\mathit{tr}(ABC+BCA+CAB)$ is equal to (where the order of matrices  $A$, $B$, $C$ is $3$ &  $tr\left(A\right)$is sum of diagonal elements $A$)

If  $p{x}^4+q{x}^3+r{x}^3+sx+t=\left|\begin{array}{ccc}{x}^2+3x& x-1& x+3\\ x+1& 2-x& x-3\\ x-3& x+4& 3x\end{array}\right|$ then $t$ is equal to -