Operation on Matrices

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

Let $A=\left[\begin{array}{lll}0& 1& 2\\ a& 0& 3\\ 1& c& 0\end{array}\right]$, where $a,c\in R$. If $A^3=A$ and the positive value of $a$ belongs to the interval $(n-1, n]$, where $n\in \mathrm{\mathbb{N}}$, then $n$ is equal to ____.

Let $\omega =\frac{1}{2}\left(-1+i\sqrt{3}\right)$ where $i^2=-1$. If $\left[\begin{array}{cc}1& \omega \\ {\omega }^2& \omega +{\omega }^2\end{array}\right]\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\left[\begin{array}{c}6\omega +2\\ 4\omega \end{array}\right]$, then $\alpha -\beta$ is equal to :

Assume that ${\left[\begin{array}{cc}0& a\\ b& 0\end{array}\right]}^4=I$. Then

Let $A$ be a $2\times 2$ matrix with real entries and $A^{\text{'}}$ its transpose such that $A^{\text{'}}=\alpha A+1,$  where $\alpha \in \mathrm{ℝ}-\left\{-1,1\right\}$., If det $\left(A^2-A\right)=4$, the sum of all possible values of $\alpha$ is equal to

Let $A=\left[\begin{array}{cc}1& \frac{1}{51}\\ 0& 1\end{array}\right]$. If $B=\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right] A\left[\begin{array}{cc}-1& -2\\ 1& 1\end{array}\right]$, then the sum of all the elements of the matrix $\sum _{n=1}^{50}{B}^n$ is equal to

Find the product of the matrices, $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\left[\begin{array}{ccc}2& 3& 4\end{array}\right]$

Let $A={\left[a_{ij}\right]}_{2\times 2},$ where $a{}_{ij}\ne 0$ for all $i,j$ and $A^2=I$, Let $\mathrm{a}$ be the sum of all diagonal elements of $A$ and $b=\left|A\right|$ Then $3a^2+4b^2$ is equal to

Let $P$ be a square matrix such that $P^2=I-P$. For $\alpha , \beta , \gamma , \delta \in \mathrm{\mathbb{N}}$, if $P^{\alpha }+P^{\beta }=\gamma l-29P$ and $P^{\alpha }-P^{\beta }=\delta l-13P$, then $\alpha +\beta +\gamma -\delta$ is equal to

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ and $Q=PA{P}^T$. If $P^T{Q}^{2007} P=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$then $2a+b-3c-4d$ is equal to

Two matrices of order $A$ and $B$ of order $p\times q$ and $r\times s$ respectively can be subtracted if