Vinod Singh and Shweta Pawar Solutions for Chapter: Matrices, Exercise 2: Critical Thinking

A square, non-singular matrix $A$ satisfies $A^2-A+2I=0$, then $A^{-1}$ is equal to

If $A$ is a square matrix such that  $\left|A\right|\ne 0$ and $A^2+mA+nI=0$, then $A^{-1}=$

If a matrix $A$ is such that $4A^3+2A^2+7A+I=0$, then $A^{-1}$ equals

The solution of the equations $5x-7y=2$ $, 7x-5y=3$ are

Let $X=\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right], A=\left[\begin{array}{lll}1& -1& 2\\ 2& 0& 1\\ 3& 2& 1\end{array}\right]$ and $B=\left[\begin{array}{l}3\\ 1\\ 4\end{array}\right]$. If $AX=B$, then $X=$

Let $X=\left[\begin{array}{l}x\\ y\\ z\end{array}\right], D=\left[\begin{array}{l}3\\ 5\\ 11\end{array}\right]$ and $A=\left[\begin{array}{rrr}1& -1& -2\\ 2& 1& 1\\ 4& -1& -2\end{array}\right]$, if $X=A^{-1}D$, then $X$ is equal to

If $n\ne 3k$ and $1, \omega , {\omega }^2$ are the cube roots of unity, then $\mathrm{\Delta }=\left|\begin{array}{ccc}1& {\omega }^n& {\omega }^{2n}\\ {\omega }^{2n}& 1& {\omega }^n\\ {\omega }^n& {\omega }^{2n}& 1\end{array}\right|$ has the value, where $k\in I$

The inverse of $\left[\begin{array}{cc}1& \sin \alpha \\ -\sin \alpha & -1\end{array}\right]$ is