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

Let $A=\left[\begin{array}{cc}2& -5\\ 3& 1\end{array}\right]$ what is $f\left(A\right),$ where $f\left(x\right)=x^3-2{\mathrm{x}}^2-5$.

The value of  $(x+y)$, where $x$ and $y$ are satisfying $\left[\begin{array}{cc}3& -2\\ 3& 0\\ 2& 4\end{array}\right]\left[\begin{array}{ll}y& y\\ x& x\end{array}\right]=\left[\begin{array}{cc}3& 3\\ 3y& 3y\\ 10& 10\end{array}\right]$ is

If $A$ and $B$ are square matrices of same order and $A{A}^{\mathit{T}}\mathit{=}I$  then ${\left(A^{\mathit{T}}BA\right)}^{10}$ is equal to -

Consider the following in respect of matrices $A$ and $B$ of same order
$1$. $A^2-B^2=(A+B)(A-B)$
$2$. $(A-I)(I+A)=O\iff A^2=I$
where $I$ is the identity matrix and $O$ is the null matrix.

Which of the above is/are correct?

Which of the following is not true?

If a square matrix $A$ is such that $\left(A^T-\frac{1}{2}I\right)\left(A-\frac{1}{2}I\right)=\left(A^T+\frac{1}{2}I\right)\left(A+\frac{1}{2}I\right)=I,$ where $I$ is a unit matrix, then $A$ is

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 $A\left(\theta \right)=\left[\begin{array}{cc}\sin \theta & i\cos \theta \\ i\cos \theta & \sin \theta \end{array}\right]$ then which of the following is not true?