Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 1: AMU-AT (B.Tech.) 2017

If $A=\left[\begin{array}{lll}0& 1& 2\\ 1& 2& 3\\ 3& \alpha & 1\end{array}\right],A^{-1}=\left[\begin{array}{ccc}\frac{1}{2}& -\frac{1}{2}& \frac{1}{2}\\ -4& 3& \beta \\ \frac{5}{2}& -\frac{3}{2}& \frac{1}{2}\end{array}\right]$ then

If $a, b, c$ are the integers between $1$ and $9$ and $a 51, b 41, c 31$ are three-digit numbers and the value of determinant $D=\left|\begin{array}{ccc}5& 4& 3\\ a51& b41& c31\\ a& b& c\end{array}\right|$ is zero, then $a, b \& c$ are

If $A=\left[\begin{array}{ll}k& l\\ m& n\end{array}\right]$ and $kn\ne lm,$ then the value of $A^2-(k+n)A+(kn-lm)I$ equals

If $A$ is an orthogonal matrix, then

If the system of equations

$2x+ay+6z=8$

$x+2y+bz=5$

$x+y+3z=4$

has a unique solution then

If $a, b, c$ are roots of the equation $x^3+px+q=0,$ then the value of $\left|\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is

If $A=\left[\begin{array}{cc}2& 3\\ 5& -2\end{array}\right]$ be such that $A^{-1}=\lambda A.$ Then the value of $\lambda$ is

If $A=\left[\begin{array}{ll}\alpha & 0\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{ll}1& 0\\ 5& 1\end{array}\right]$ and $A^2=B,$ then the value of $\alpha$ is