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

If $A$ is a $3-$rowed square matrix and $\left|3A\right|=k\left|A\right|$ then $k=$?

If $A$ is a square matrix of order $3$ and $\left|A\right|=5$ then the value of $\left|2A^{\text{'}}\right|$ is 

$\left|\begin{array}{ccc}\sin \theta & \cos \theta & \sin 2\theta \\ \sin \left(\theta +\frac{2\mathrm{\pi }}{3}\right)& \cos \left(\theta +\frac{2\mathrm{\pi }}{3}\right)& \sin \left(2\theta +\frac{4\mathrm{\pi }}{3}\right)\\ \sin \left(\theta -\frac{2\mathrm{\pi }}{3}\right)& \cos \left(\theta -\frac{2\mathrm{\pi }}{3}\right)& \sin \left(2\theta -\frac{4\mathrm{\pi }}{3}\right)\end{array}\right|=$

If $A$ is a $2$-rowed matrix and $\left|A\right|=6$ then $A$. $adjA=$? 

If $A=\left[\begin{array}{r}3\hspace{0.25em} \hspace{0.25em}4 \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1 \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em} 0\hspace{0.25em}\hspace{0.25em}-2\\ -2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1 \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\end{array}\right]$ then $A^{-1}=?$

If $A$ is an invertible matrix and $A^{-1}=\left[\begin{array}{cc}3& 4\\ 5& 6\end{array}\right]$ then $A=?$

Let $A$ be a non-singular square matrix of order $3\times 3.$ Then $\left|adjA\right|$ is equal to:

If $x, y, z$ are non-zero real numbers, then the inverse of the matrix $A=\left[\begin{array}{lll}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$ is