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

Let $A$ and $B$ be $3\times 3$ symmetric matrices such that $X=AB+BA$ and $Y=AB-BA$. Then $[{\left(XY\right)}^{\mathrm{T}}$ is equal to ${\left(XY\right)}^T$ is the transpose of matrix $XY.]$

Let $A \& B$ be $3\times 3$ symmetric matrices such that  $X=AB+BA$ and $Y=AB-BA$. Then $(XY{)}^T$ is equal to $\left[\right(XY{)}^T$ is the transpose of matrix $\mathrm{XY}.]$

If $A$ and $B$ are non-singular square matrices of even order such that $A^T=A, B^T=-B$ and $AB+BA=O$

(where $O$ is a null matrix ), then

$A$ is a $3\times 3$ matrix with entries from the set $\left\{-1,0,1\right\}$. Then the probability that $A$ is neither symmetric nor skew-symmetric is

If $A$ is a skew symmetric matrix of order $2$ and $B,C$ are matrices $\left[\begin{array}{cc}1& 4\\ 2& 9\end{array}\right], \left[\begin{array}{cc}9& -4\\ -2& 1\end{array}\right],$ then $A^{3}BC+A^5\left(B^2{C}^2\right)+A^7\left(B^3{C}^3\right)+\dots ..A^{2017}\left(B^{1009}{C}^{1009}\right)$ is

If $A$ is skew symmetric matrix such that $A^2-I=0$ then order of $A$ is :-

If $A$ is skew symmetric matrix such that $A^2-I=0$ then order of $A$ is :-

Let $f\left(x\right)=\mathcal{l}n\left(x+\sqrt{x^2+1}\right)$, then the value of determinant $\left|\begin{array}{lll}f\left(\sin 2017\pi \right)& f\left(\sin \frac{\pi }{6}\right)& f\left(e^x\right)\\ f\left(\cos \frac{2\pi }{3}\right)& f\left(\cos \frac{2017\pi }{2}\right)& f\left(\tan \frac{\pi }{3}\right)\\ f\left(-e^{\mathrm{x}}\right)& f\left(\cot \frac{5\pi }{6}\right)& f\left(0\right)\end{array}\right|$ is

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

Which of the following option is correct for $X^m$, if $\mathrm{X}$ is a symmetric matrix and $\mathrm{m}$ is any natural number.

What type of matrix would $\frac{1}{2}(\mathrm{X}-\mathrm{X}\text{'})$ be, if $\mathrm{X}$ is a square matrix and ${\mathrm{X}}^{\text{'}}$ is the transpose of $\mathrm{X}$.

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?

The trace of a $5\times 5$ skew symmetric matrix is equal to:

If $A$ and $B$ are any two $3\times 3$ matrices such that $A$ is a symmetric and $B$ is a skew symmetric matrix, then the matrix $AB-BA$ is

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