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 $p$ is a $5\times 5$ skew-symmetric matrix with $9$ entries as $1,$ other $9$ entries as $-1$ and $7$ entries as $0$. If $tr\left(p\right)$ is sum of diagonal elements, then ${\left(tr\left(p\right)\right)}^2-det\left(p\right)$ is equal to

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 $x$.

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