Tag: properties of matrix multiplication

If $ \begin{bmatrix}
              2 & 1 \[0.3em]
              3 & 2
              \end{bmatrix} \ A \begin{bmatrix}
              -3 & 2 \[0.3em]
              5 & -3
              \end{bmatrix} = \begin{bmatrix}
              1 & 0 \[0.3em]
              0 & 1
              \end{bmatrix}$ then the matrix A is equal to

Find the number of all possible ordered sets of two $(n\times n)$ matrices A and B for which $AB-BA=$$I$.

If $A=|a _{ij}| _{2\times 2}$, where $a _{ij}=\left{\begin{matrix} i+j, & if & i\neq j\ i^2-2j, & if & i=j\end{matrix}\right.$, then $A^{-1}=?$

If $\omega$ is the complex cube root of unity, then inverse of $\begin{bmatrix} \omega  & 0 & 0 \ 0 & { \omega  }^{ 2 } & 0 \ 0 & 0 & { \omega  }^{ 2 } \end{bmatrix}$ is

The inverse of the matrix $\begin{bmatrix}1 & 0 & 1\ 0 & 2 & 3\ 1 & 2& 1\end{bmatrix}$ is

If A =$\left[ \begin{matrix} i \ 0 \end{matrix}\begin{matrix} 0 \ -1 \end{matrix} \right] $, than check whether: ${{\text{A}}^2} =  - {\text{I,(}}{{\text{i}}^2} =  - 1)$

If $M = \left[ \begin{array}{l}0\,\,\,\,2\5\,\,\,\,\,0\end{array} \right]\,\,\,and\,\,N = \left[ \begin{array}{l}0\,\,\,\,5\2\,\,\,\,\,0\end{array} \right]$,then ${M^{2011}}$ is-

If $A = \left[ \begin{array}{l}\cos \theta \,\,\,\,\sin \theta \ - \sin \theta \,\,\,\cos \theta \end{array} \right]$ where $\theta  = \frac{{2\pi }}{{19}}$ then ${A^{2017}} = $

If A and B are matrices of the same order, then $\displaystyle :\left ( A+B \right )^{2}= A^{2}+2AB+B^{2}$ is possible, iff

If $A$ and $B$ are any two matices, then