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

Find the non-zero roots of the equation,

$\left|\begin{array}{ccc}15-2x& 11& 10\\ 11-3x& 17& 16\\ 7-x& 14& 13\end{array}\right|=0$

Find the value of $\left|\left(adj\left(adj\left(adjA\right)\right)\right)\right|$ in terms of $\left|A\right|$

If $A$ is a symmetric and $B$ a skew-symmetric matrix and $\left(A+B\right)$ is non-singular and $C={\left(A+B\right)}^{-1}\left(A-B\right)$ then prove that
 $C^{\top }\left(A-B\right)C=A-B$.

Apply Cramer's rule to solve the following simultaneous equations.
$x+2 y+3 z=2$
$x-y+z=3$
$5 x-11 y+z=17$

Solve the following systems of linear equations by matrix method.

$x+y+z=9$

$2 x+5 y+7 z=52$

$2 x+y-z=0$

A matrix ${}^{\text{'}}{A}^{\text{'}}$ has $6$ elements. Find the number of possible orders of $A$.

Prove the statement: For any $2\times 2$ matrix $A,$ if $A\left(adjA\right)=\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right],$ then $\left|A\right|=10$

Prove that if $A$ is skew symmetric, then $B^{\text{'}}AB$ is also skew symmetric.