Symmetric and Skew-symmetric Matrices

$A=\left[\begin{array}{cc}2& 0\\ 3& 5\end{array}\right]$, express as $P+Q$, where $P$ is symmetric and $Q$ is skew symmetric.

The matrix $A=\left[\begin{array}{ccc}6& 8& 5\\ 4& 2& 3\\ 9& 7& 1\end{array}\right]$ is the sum of symmetric matrix B and skew symmetric matrix C. Find C.

A square matrix is a skew-symmetric matrix if 

If $A=\left[\begin{array}{cc}0& x+2\\ 2x-3& 0\end{array}\right]$ is skew-symmetric matrix then the value of $x$ is

If $A=\left[a_{ii}\right]$ is a skew-symmetric matrix of order $n$ then $\sum _{}{a}_{ii}=$

If $A=\left[a_{ii}\right]$ is a skew symmetric matrix of order $n$ then $\sum _{}^{}{a}_{ii}=$

$A=\left[\begin{array}{ccc}3& a& -1\\ 2& 5& c\\ b& 8& 2\end{array}\right]$ is symmetric and $B=\left[\begin{array}{ccc}d& 3& a\\ b-a& e& -2b-c\\ -2& 6& -f\end{array}\right]$ is skew-symmetric, then find $AB.$ 

If $A=\left[\begin{array}{cc}\alpha & a\\ \beta & b\\ \gamma & c\end{array}\right]$, then $A{A}^T$ is

If $A$ and $B$ are symmetric matrices of the same order, then show that $AB$ is symmetric if and only if $A$ and $B$ commute, that is $AB=BA$.

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.

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

Which function of money eliminates the need for a double coincidence of wants?