Algebra of Matrices

If matrix A = $\left[\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& 1-\text{a}\end{array}\right]$ is such that A² = A. when $\text{bc}=1/4$ then a is equal to

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 value of $x+y$ from the following equation is $2\left(\begin{array}{c}1\\ 0\end{array}\begin{array}{c}3\\ x\end{array}\right)+\left(\begin{array}{c}y\\ 1\end{array}\begin{array}{c}0\\ 2\end{array}\right)=\left(\begin{array}{c}5\\ 1\end{array}\begin{array}{c}6\\ 8\end{array}\right)$.

If $M=\left(\begin{array}{cc}\frac{5}{2}& \frac{3}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$ ?

Find the inverse of the following matrix:

$A=\left[\begin{array}{ccc}3& 2& 1\\ 4& -1& 2\\ 7& 3& -3\end{array}\right]$

For two matrices $A \mathrm{and} B$, $AB=0$, If and only if

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

If $A=\left[\begin{array}{cc}5& 2016\\ 1& 403\end{array}\right]$, then the value of $det\left(A^{2015}-5A^{2014}\right)=$________

Let $A$ be a $4\times 4$ matrix with real entries. Consider the sets

$K=\left\{\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right): A\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right)\right\},$

$J=\left\{\left(\begin{array}{c}{c}_1\\ c_2\\ c_3\\ c_4\end{array}\right): \left(\begin{array}{c}{c}_1\\ c_2\\ c_3\\ c_4\end{array}\right)=A\left(\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\end{array}\right)\right\}$ for some $\left.y_1,y_2,y_3.y_4\in R\right\}$

Suppose $K=J$. Then, which one of the following statements is necessarily true?

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