Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 1: Assam CEE 2016

If $A$ is a $3\times 3$ non-singular matrix such that $A{A}^T=A^{T}A$ and $B=A^{-1}{A}^T$, then $B{B}^T$ is equal to:

If $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right], 10B=\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& \alpha \\ 1& -2& 3\end{array}\right]$ and $B$ is the inverse of matrix $A$, then $\alpha$ is:

If $A$ and $B$ are symmetric matrices of same order and $X=AB+BA$ and $Y=AB-BA$, then ${\left(XY\right)}^T$ is equal to

If for a square matrix $A$ of even order, $A^T=-A$, then ${\left(A^{-1}\right)}^T$ is equal to

If the system of equations $x=a\left(y+z\right),y=b\left(z+x\right),z=c\left(x+y\right);a,b,c\ne -1$ has a non trivial solution, then the value of $\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}$ is

What is the possible number of different orders, which a matrix can have when it has $24$ entries?

$\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$ if

The value of $f\left(100\right)$ is, if

$f\left(x\right)=\left|\begin{array}{ccc}1& x& \left(x+1\right)\\ 2x& x\left(x-1\right)& \left(x+1\right)x\\ 3x\left(x-1\right)& x\left(x-1\right)\left(x-2\right)& \left(x+1\right)x\left(x-1\right)\end{array}\right|$