Symmetric and Skew-Symmetric Matrices

Using matrices, solve the following system of equations

 $\begin{array}{l}4x-5y-11z=12\end{array}$

$x-3y+z=1$

$2x+3y-7z=2$

Let $\left|M\right|$ denote the determinant of a square matrix $M$. Let $g:\left[0,\frac{\pi }{2}\right]\to \mathrm{ℝ}$ be the function defined by $g\left(\theta \right)=\sqrt{f\left(\theta \right)-1}+\sqrt{f\left(\frac{\pi }{2}-\theta \right)-1}$ where $f\left(\theta \right)=\frac{1}{2}\left|\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta +\frac{\pi }{4}\right)& \tan \left(\theta -\frac{\pi }{4}\right)\\ \sin \left(\theta -\frac{\pi }{4}\right)& -\cos \frac{\pi }{2}& {\log }_e\left(\frac{4}{\pi }\right)\\ \cot \left(\theta +\frac{\pi }{4}\right)& {\log }_e\left(\frac{\pi }{4}\right)& \tan \pi \end{array}\right|$
Let $p\left(x\right)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g\left(\theta \right)$, and $p\left(2\right)=2-\sqrt{2}$. Then, which of the following is/are TRUE ?

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

Let $A$ be set of all $3\times 3$ symmetric matrices all of whose entries are either $0$ or $1.$ Five of these entries are $1$ and four of them are $0$, Then

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

If $A=\left[\begin{array}{ccc}-2& -1& 1\\ -1& 7& 4\\ 1& -x& -3\end{array}\right]$ is a symmetric matrix then the value of $x$ 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 $A=\left[\begin{array}{ccc}-2& -1& 1\\ -1& 7& 4\\ 1& -x& -3\end{array}\right]$ is a symmetric matrix then find the value of $x$.

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

If $A=\left[\begin{array}{ccc}2& 1& 1\\ -1& 0& 2\\ 0& 1& 3\end{array}\right]$ then prove that $A{A}^T$ and $A^{T}A$ are symmetric matrix.

If $A=\left[\begin{array}{ccc}2& 1& 1\\ -1& 0& 2\\ 0& 1& 3\end{array}\right]$ then prove that $A-A^T$ is a skew-symmetric matrix.

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

Express the matrix $A$ as the sum of symmetric and skew-symmetric matrix where $A=\left[\begin{array}{cc}6& 2\\ 5& 4\end{array}\right]$

If the matrix $\left[\begin{array}{ccc}0& a& 3\\ 2& b& -1\\ c& 1& 0\end{array}\right]$ is a skew symmetric matrix, then find the values of $a+b+c$

If $A$ and $B$ are two skew symmetric matrices of same order, then $AB$ is symmetric matrix if _____

Show that a matrix which is both symmetric and skew symmetric is a zero matrix.

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?