Embibe Experts Solutions for Exercise 4: Assignment

If $A$ is a square matrix, then $A$ is skew-symmetric, iff

The matrix $X$ in the equation $AX=B$, such that $A=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$ is given by

Let $A={\left[a_{ij}\right]}_{50\times 50}$ such that $a_{ij}={\tan }^{-1}\left(i-j\right),\forall i, j$, then $\sum a_{ij}$, is

Let a set $A=\left\{a_1,a_2,a_3,\dots ,a_{180}\right\}$. Then total number of possible matrices of different order formed by using all element of set $A$ exactly once in a matrix is $n_1·\left(n_2!\right)$, where $n_1,n_2\in N$. Then least value of $\left(n_1+n_2\right)$ is

$A$ and $B$ are two $3\times 3$ matrices such that they are inverse of each other then $tr.\left(5AB+6BA+7{\left(AB\right)}^2+8{\left(BA\right)}^2\right)$ is equal to

If $A$ and $B$ are orthogonal matrices of order $4\times 4$ and $5\times 5$ respectively then $tr.\left(A{A}^T+2A^{T}A+5\left(A{A}^T\right)+tr.\left(2B{B}^T+4B^{T}B\right)\right)$ is

Let $A=\left[\begin{array}{cc}3& 1\\ -16& -5\end{array}\right]$, then the sum of all the elements of $A^{50}$ is equal to

Let $A=\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$ and $B=A+A^2+A^4+A^8+....+A^{2^{78}}$ then sum of all the elements of matrix $B$ is equal to