Matrices

Let $A$and $B$be two$3\times\,3$non-zero real matrices such that $AB$is a zero matrix. Then

The positive value of the determinant of the matrix$A$, whose $Adj\left(Adj\left(A\right)\right)=\left[\begin{matrix} 14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14 \end{matrix}\right]$, is ______?

Let$A=\left(\begin{matrix} 2 & -1 \\ 0 & 2 \end{matrix}\right)$. If$B=I-C15\left(\operatorname{adj}\,A\right)+C25(\operatorname{adj}\,A)^{2}-...-C55(\mathrm{adj}A)^{5}$,then the sum of allelements of the matrix $B$ is:

Let $M$ be any $3\times\,3$matrix with entries from the set $\left{0,1,2\right}$.The maximum number of such matrices, for which the sum of diagonal elements of $M^{T}M$ is seven, is______.

Let $A=\left[\begin{matrix} a_{1} \\ a_{2} \end{matrix}\right]$ and $B=\left[\begin{matrix} b_{1} \\ b_{2} \end{matrix}\right]$ be two $2\times\,1$matrices with real entries such that $A=XB,$where…

The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $\mathrm{A}^{\mathrm{T}} \mathrm{A}$ is 5, is

Let $A=\left[\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\right].$Then the number of $3\times\,3$ matrices $B$ with entries from the set ${1,2,3,4,5}$ and satisfying $AB=BA$ is ________.

Let $P=\left[\begin{matrix} 3 & -1 & -2 \\ 2 & 0 & \alpha\, \\ 3 & -5 & 0 \end{matrix}\right],$where $\alpha\,\in\,R.$ Suppose $Q=\left[q_{ij}\right]$ is a matrix satisfying $PQ=kI_{3}$ for some non-zero $k\in\,R.$If…

Let $A=\left[\begin{array}{ccc}0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0\end{array}\right]$ and $B$ be a matrix such that $B(I-A)=I+A$. Then the sum of the diagonal elements of $\mathrm{B}^{\mathrm{T}} \mathrm{B}$ is equal…

Let $A$ be a $2\times\,2$ real matrix and $I$ be the identity matrix of order $2.$ If the roots of the equation $|A-\mathrm{xI}|=0$ be $-1$ and $3,$then the sum of the diagonal elements of the matrix $A^{2}$is _____.

Let for $A=\left[\begin{matrix} 1 & 2 & 3 \\ \alpha\, & 3 & 1 \\ 1 & 1 & 2 \end{matrix}\right],\left|A\right|=2$. If $|2\operatorname{adj}\,(2\operatorname{adj}\,(2A))|=32^{n}$, then $3n+\alpha\,$ is equal to

Let $A=\left[\begin{matrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}\right],B=\left[\begin{matrix} B_{1} & B_{2} & B_{3} \end{matrix}\right]$, where $B_{1}$, $B_{2},B_{3}$are column matrices, and…