Tag: properties of multiplication of matrix

If $A=\begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1 \end{bmatrix}$ then $A^n=\begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \ 3^{n-1} & 3^{n-1} & 3^{n-1} \ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix}$ , $n \in N$

If $A = \begin{bmatrix}1\ 2\ 3
\end{bmatrix}$ then $AA^{1}$.

If f (x)= $\left[ {\begin{array}{{20}{c}}  {\cos \,x}&{ - \sin \,x}&0 \   {\sin \,x}&{\cos \,x\,}&0 \   0&0&1 \end{array}} \right]$ and $\left[ {\begin{array}{{20}{c}}  {\cos \,x}&0&{\sin \,x} \   0&1&0 \   { - \sin \,x}&0&{\cos \,x} \end{array}} \right]$
then ${\left[ {f\left( x \right)g\left( y \right)} \right]^{ - 1}}$ is equal to 

If for the matrix $A.A^3=1$, then $A^{-1}=$

Let $A$ be a square matrix such that $A^2 = A$ and $|A| \neq 0$, then choose the correct option.

For two matrices $A$ and $B$, if $AB=0$, then

For any non-singular matrix A, $ \displaystyle A^{-1} $ =

If $A=\begin{bmatrix} \cos { \alpha  }  & -\sin { \alpha  }  \ \sin { \alpha  }  & \cos { \alpha  }  \end{bmatrix}$, $B=\begin{bmatrix} \cos { 2\beta  }  & \sin { 2\beta  }  \ \sin { 2\beta  }  & -\cos { 2\beta  }  \end{bmatrix}$, where 0 < $\beta$ < ${ \pi  }/{ 2 }$, then prove that $BAB=$ ${ A }^{ -1 }$.

Let $A$ be a $3\times 2$ matrix with real entries. Let $H = A(A^{T}A)^{-1}A^{T}$ where $A^{T}$ is the transpose of $A$ and let $I$ be the identity matrix of order $3\times 3$. Then

If $A^3 = 0$ then $1 + A + A^2$ is equal to