Symmetric and Skew-symmetric Matrices

The number of symmetric matrices of order 3, with all the entries from the set $\{0,1,2,3,4,5,6,7,8,9\}$ is

A square matrix is a skew-symmetric matrix if 

If $A$ is a skew symmetric matrix, then $A^{2021}$ is

Let $\mathrm{S}=\left\{\left[\begin{array}{c}\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\end{array}\right]: a_{ij}\in \left\{-1,0,1\right\}\right\}$, then the number of symmetric matrices with trace equals zero, is

What are the maximum number of distinct entries that are possible in a Skew Symmetric Matrix of Order $n$?

For $∆ABC$, the value of $\left|\begin{array}{ccc}0& \sin A& \tan B\\ -\sin \left(B+C\right)& 0& \cos C\\ \tan \left(A+C\right)& -\cos C& 0\end{array}\right|=\_\_\_\_\_$.

Let $P$  be a $3\times 3$ real matrix. Let $Q=\frac{1}{2}\left(P+P^T\right)$ where $P^T$ is the transpose of $P$. Then

Which one is an example of Symmetric Matrix ?

Let $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$ whose entries are real numbers. Which of the following statements is ALWAYS TRUE?

Let $A$ and $B$ be $3\times 3$ matrices. Consider the following statements:
(I) If $A$ and $B$ are diagonal, then so is $AB$
(II) If $A$ and $B$ are symmetric, then so is $AB$

Let $A, B$ be invertible $2\times 2$ matrices such that $A$ is symmetric and $B$ is skew-symmetric. Let $C=A^{-1}{B}^{-1}+B^{-1}{A}^{-1}$ and $D=B^{-1}AB.$ Then

If $A$ and $B$ are matrices of same order, then $\left(A{B}^{\text{'}}-B{A}^{\text{'}}\right)$ is a

If $A$ is a square matrix then $\left(A+A^{\text{'}}\right)$ is

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

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?