Embibe Experts Solutions for Chapter: Matrices, Exercise 1: Manipur Board-2018

Define identity matrix.

If $A$ is a matrix of order $2\times 3$ and $B$ is a matrix such that $A\text{'}B$ and $BA\text{'}$ both are defined, then what is the order of $B$?

Prove that every square matrix is uniquely expressible as the sum of a symmetric matrix and a skew-symmetric matrix.

If $A=\left[\begin{array}{ccc}2& 1& 1\\ 2& 0& 1\\ 0& -2& -1\end{array}\right]$, find $A^{-1}$. Using $A^{-1}$ solve the system of linear equations: 

$2x+y+z=3, 2x+z=5,-2y-z=1$.

If $A=\left[\begin{array}{cc}0& -\tan \frac{\alpha }{2}\\ \tan \frac{\alpha }{2}& 0\end{array}\right]$ and $I$ is the identity matrix of order $2$, show that  $I+A=(I-A)\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$.

If the inverse of a square matrix exists, prove that it is unique. If $A$ and $B$ are both invertible square matrices of the same order, prove that $(AB{)}^{-1}=B^{-1}{A}^{-1}$.

For any square matrix $A$, prove that $AA\text{'}$ is a symmetric matrix.

If $A=\left[\begin{array}{ccc}1& 0& -2\\ -2& -1& 2\\ 3& 4& 1\end{array}\right]$, show that $A^3-A^2-3A-I=O$ and hence find $A^{-1}$.