Operation on 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 $\left[\begin{array}{cc}3& -2\\ 3& 0\\ 2& 4\end{array}\right]\left[\begin{array}{cc}y& y\\ x& x\end{array}\right]=\left[\begin{array}{cc}3& 3\\ 3y& 3y\\ 10& 10\end{array}\right]$

If $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]; 10B=\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& a\\ 1& -2& 3\end{array}\right]; \text{and} B=A^{-1}, \text{then the value of} a is?$

$If \mathrm{A}= \left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$ Then find the value of ${\mathrm{A}}^2$

Matrix A has $x$ Rows and $x+5$ Columns, Matrix B has $y$ Rows and  $11-y$ coloums.If both AB and BA exist, then the respective values of $x$ and $y$ are

If $A$ and $B$ are two matrices such that $AB=A \& BA=B$, then number of elements in the set $\left\{A=\left[\begin{array}{cc}a& b\\ 0& d\end{array}\right]: a,b,d\in \left\{-1,0,1\right\}\right\}$ is

If $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$, $P=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ and $X=AP{A}^T$ then $A^T{X}^{50}A=$

If $A=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]$ then the value of $A^5$ is

If $p,q,r\in R, \left[p q r\right]\left[\begin{array}{ccc}3& 4& 1\\ 3& 2& 3\\ 2& 0& 2\end{array}\right]=\left[3 0 1\right]$ then $2p+q-r$ ______

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?