Symmetric and Skew-Symmetric Matrices

Let $A$ and $B$ be symmetric matrices of same order, then which one of the following is correct regarding $\left(AB-BA\right)$?

$1.$ Its diagonal entries are equal but nonzero

$2.$ The sum of its non-diagonal entries is zero

Select the correct answer using the code given below :

If $A=\left[\begin{array}{ccc}2& 1& 1\\ -1& 0& 2\\ 0& 1& 3\end{array}\right]$ then $A-A^T$ is

If $x, y$ are any two non-zero real numbers, $a_{ij}=xj+yi,A={\left(a_{ij}\right)}_{n\times n}$ and $P, Q$ are two $n\times n$ matrices such that $A=x P+y Q$ then

Determinant of skew-symmetric matrix of order "three" is always

If the matrix $A$ is both symmetric and skew symmetric, then

If $A$ and $B$ are non singular square matrices of even order such that $A^{\top }=A$ and $B^{\top }=-B$ and $AB+BA=O$ (where $O$ is null matrix), then choose appropriate option

Let $A$ be the set of all $4\times 4$ skew symmetric matrices, whose entries are $-1,0$ or $1.$ If there are exactly four $0\text{'}s$, six $1\text{'}s$and six $-1\text{'}s$, then what will be the number of such matrices?

If a square matrix $A=\left[\mathrm{ }\begin{array}{lll}\mathrm{ 2}& 0& –3\\ \mathrm{ 4}& 3& \mathrm{ 1}\\ –5& 7& \mathrm{ 2}\end{array}\mathrm{ }\right]$ is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix will be

The symmetric part of the matrix $\left[\begin{array}{ccc}1& 2& 6\\ 4& 3& -4\\ 8& 6& 5\end{array}\right]$ is

If a matrix $\mathrm{A}$ is both symmetric and skew symmetric then

If $A$ and $B$ are two skew symmetric matrices of order $n$ then

If $\mathrm{A}=\left[\begin{array}{ccc}\mathrm{x}& 3& 2\\ -3& \mathrm{y}& -7\\ -2& 7& 0\end{array}\right]\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{A}=-{\mathrm{A}}^{\mathrm{\prime }},\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{ }\mathrm{x}+\mathrm{y}\mathrm{ }$ is equal to

The symmetric part of the matrix $A=\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]$ is

Let $p$ be an odd prime number and $T_P$ be the following set of $2\times 2$ matrices $T_P=\left\{A= \left[\begin{array}{cc}a& b\\ c& a\end{array}\right];a, b, c \in \left\{0,1, 2,\dots \left(p-1\right)\right\}\right\}$

The number of $A$ in $T_P$ such that $A$ is either symmetric or skew - symmetric or both, and $det\left(A\right)$ is divisible by $p$ is

If $A=\left[\begin{array}{cc}3& x-1\\ 2x+3& x+2\end{array}\right]$ is a symmetric matrix, then x =

If $A=\left[\begin{array}{cc}5& x\\ y& 0\end{array}\right]$ and $A=A^T,$ then

In a symmetric matrix of order $\mathrm{‘}12\mathrm{’}$ maximum number of distinct elements are -

If $A=\left[\begin{array}{ccc}2& 0& -3\\ 4& 3& 1\\ -5& 7& 2\end{array}\right]$ is expressed as the sum of a symmetric and skew-symmetric matrix then the symmetric matrix is 

If $\mathrm{A}$ is a skew-symmetric matrix, then trace of $\mathrm{A}$ is

A skew symmetric matrix $S$ satisfies the relation $S^2+I=0$, where $I$ is a unit matrix, then $S{S}^{\text{'}}$ is (where $S^{\text{'}}$ is transpose of $S$)