Operation on Matrices

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

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

Find $xyz$ if $A=\left[\begin{array}{ccc}0& 2y& z\\ x& y& -z\\ x& -y& z\end{array}\right]$. Given that $A\text{'}A=I_3$.

Show that $\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$ is a solution of the matrix equation ${\mathrm{X}}^2-2\mathrm{X}-3\mathrm{I}=\mathrm{O}$ where $\mathrm{I}$ is the unit matrix of order $2.$

Find the value of $\left[\begin{array}{ll}1& 3\\ 4& 0\\ 2& 1\end{array}\right]\otimes \left[\begin{array}{c}1\\ 5\end{array}\right]=\left[\begin{array}{c}?\\ ?\\ ?\end{array}\right]$

Let $\mathrm{X}=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right] \mathrm{and} \mathrm{A}=\left[\begin{array}{ccc}-1& 2& 3\\ 0& 1& 6\\ 0& 0& -1\end{array}\right]$. For $\mathrm{k}\in \mathrm{N}, \mathrm{if} \mathrm{X}\text{'}{\mathrm{A}}^{\mathrm{K}}\mathrm{X}=33,$then $\mathrm{k}$=

$\mathrm{P} \mathrm{and} \mathrm{Q}$ are matrices such that both $\left(\mathrm{P}+\mathrm{Q}\right)$ and $\left(\mathrm{PQ}\right)$ are defined.

Which of the following is true about $\mathrm{P} \mathrm{and} \mathrm{Q}$?

Find the integers p and q if $\left[\begin{array}{ll}p& q\end{array}\right]\left[\begin{array}{c}p\\ q\end{array}\right]=\left[25\right]$

If $A=\left[\begin{array}{ccc}0& a& b\\ -a& 0& c\\ -b& -c& 0\end{array}\right]\text{ and} a^2+b^2+c^2=1$ then prove that $A^3+A=O$

If $A=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right],$ find $A^2.$

If $A_n$ is a n×n matrix in which the diagonal elements are 1,2,3.....n i.e $a_{11}=1,a_{22}=2,.......a_{ii}=i,......$and all other elements are equal to n, then

$1+\frac{3}{2}+\frac{5}{2^2}+\frac{7}{2^3}+\dots \mathrm{\infty }$ is equal to 

If $A=\left(\begin{array}{ccc}1& -2& 1\\ 0& 1& -1\\ 3& -1& 1\end{array}\right)$ then find $A^3-3A^2-A-3I$

If $A=\left[\begin{array}{cc}\sin 30°+\cos 60°& \tan 45°-\cot 45°\\ \cos 90°& \sin 90°\end{array}\right]$, then prove that $A^3=A^2=A$

The value of $x$ if $\left[\begin{array}{ccc}x& \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ \sin \theta \\ \cos \theta \end{array}\right]=\left[10\right]$ is 

Find $A^T$ of $A=\left[\begin{array}{cc}-1& 5\\ 3& -2\\ 4& 7\end{array}\right]$

If $A=\left[\begin{array}{cc}1& 2\\ 2& -1\end{array}\right] \text{and} B=\left[\begin{array}{cc}0& 2\\ 3& -1\end{array}\right]$ find $AB$.

What is rank in Matrix?

If $A=\left[\begin{array}{cc}\cos \alpha & -2\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$ find $A+A$