Algebra of Matrices

Find the value of $x$ and $y$ that satisfies the equations $\left[\begin{array}{cc}3& -2\\ 3& 0\\ 2& 4\end{array}\right]\left[\begin{array}{cc}y& y\\ x& x\end{array}\right]=\left[\begin{array}{cc}3& 3\\ 3y& 3y\\ 10& 10\end{array}\right]$

The matrix $X$ is such that  $2A+B+X=0$ , where $A=\left[\begin{array}{cc}-1& 2\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}3& -2\\ 1& 5\end{array}\right]$. The matrix $X$ is:

Show that for two matrices A and $B$, $\mathrm{tr}\left(\mathrm{AB}\right)\ne \mathrm{tr}\left(\mathrm{A}\right)\mathrm{tr}\left(\mathrm{B}\right)$

Trace of an identity matrix with order $n\times n$ is

Check weather given matrices of the order $3\times 3$ and $2\times 2$ are to be comparable or not?

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$. Verify closure property of matrix multiplication with scalar $3$.

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$. Verify closure property of matrix multiplication with scalar $2$.

$A=\left[\begin{array}{cc}6& 7\\ 5& 2\end{array}\right]$. Prove distributive property of scalar multiplication of a matrix with scalars $2$ and $3$.

$A=\left[\begin{array}{cc}1& 4\\ 3& 5\end{array}\right]$. Prove distributive property of scalar multiplication of a matrix with scalars $2$ and $3$.

$A=\left[\begin{array}{cc}1& 2\\ 4& 6\end{array}\right]$. Prove distributive property of scalar multiplication of a matrix with scalars $2$ and $3$.

$A=\left[\begin{array}{cc}1& 4\\ 3& 5\end{array}\right]$. Find additive inverse of $A$.

$A=\left[\begin{array}{cc}2& 1\\ 4& 6\end{array}\right]$. Find additive inverse of $A$.

If $A=\left[\begin{array}{cc}5& 7\\ 6& 8\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$. Prove the closure property of addition.

If $A=\left[\begin{array}{cc}2& 3\\ 8& 6\end{array}\right]$, $B=\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]$. Prove the closure property of addition.

If $A=\left[\begin{array}{cc}1& 7\\ 2& 3\end{array}\right]$, $B=\left[\begin{array}{cc}3& 6\\ 5& 8\end{array}\right]$. Prove the closure property of addition.

If $A=\left[\begin{array}{cc}1& 2\\ 4& 6\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ 5& 3\end{array}\right]$. Prove the closure property of addition.

If $A=\left[\begin{array}{cc}4& 5\\ 1& 2\end{array}\right]$ and $C_i, R_i$ represents $i^{th}$ column and row respectively, find new matrix obtained for $2R_1, 3R_2$.