Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 1: TS EAMCET 2018 (04-May Shift-1)

If $A_{\alpha }=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$, then determinant of $A_{\pi /5}{A}_{\pi /4}{A}_{3\pi /10}=$

Let $A$ and $B$ be two $3\times 3$ non singular matrices, such that det $\left(A^{T}BA\right)=27$ and det $\left(A{B}^{-1}\right)=8.$ Then det $\left(B^T{A}^{-1}B\right)=$

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

Let $B=\left[\begin{array}{ll}1& 2\\ 0& 1\end{array}\right]$ and $A$ be a $2\times 2$ matrix satisfying ${\left(A^T\right)}^{-1}=A.$

If $X=AB{A}^T,$ then $A^T{X}^{2021}A=$

If $A=\left(\begin{array}{lll}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right),$ then $adj(adjA)$ is equal to

If $A$ is a $3\times 3$ matrix such that $\left|A\right|=27, Adj\left(A\right)=k{A}^T,$ then $k^2-3k+5=$

If $x=\alpha , y=\beta , z=y$ is the unique solution of the system of equations $5x-7y+3z=0,$ $7x+10y-8z=3$ and $2x+3y-4z+4=0,$ then $\beta =$

If the system of equations $2x+9y+5z=8, 2x+3y-z=-4, x-2z=-5$ have an infinite number of solutions $x=-5+at, y=2+bt, z=ct, t\in R$ then $a, b, c$ respectively are