Matrices

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

Let$P=\left[\begin{matrix} -30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{matrix}\right]$ and $A=\left[\begin{matrix} 2 & 7 & \omega\,^{2} \\ -1 & -\omega\, & 1 \\ 0 & -\omega\, & -\omega\,+1…

Let $A = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ satisfy $A^2 + \alpha(adj(adj(A))) + \beta(adj(A)(adj(adj(A)))) = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}$ for…

The number of matrices $A=\left[\begin{matrix} a & b \\ c & d \end{matrix}\right]$, where $a,b,c,d\in\,\left{-1,0,1,2,3,\ldots\,\ldots\,,10\right}$, such that $A=A^{-1}$, is ______?

Let $A=\left{a_{ij}\right}$be a $3\times\,3$ matrix, where$a_{ij}=\left{\begin{matrix} \left(-1\right)^{j-i}\mathrm{if}ij \end{matrix}\right$then $\det\,\left(3\operatorname{Adj}\,\left(2A^{-1}\right)\right)$ is equal…

If$A=\left[\begin{matrix} 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3 \end{matrix}\right], B=adjA$ and $C=3A,$ then $\frac{\left|adjB\right|}{\left|C\right|}$ is equal to

Consider the matrices $A = \begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 9 \\ 1 & 3 \end{bmatrix}$. If matrices P and Q are such that $PA = B$ and $AQ = B$, then the absolute value of the…

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 $x,y,z>1$and$A=\left[\begin{matrix} 1 & \log\,_{x}y & \log\,_{x}z \\ \log\,_{y}x & 2 & \log\,_{y}z \\ \log\,_{z}x & \log\,_{z}y & 3 \end{matrix}\right]$.Then…

Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $\mathrm{A}=\left[\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right],|\mathrm{A}|=-1$. Let B be the inverse of the…

If $A=\left[\begin{matrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{matrix}\right],B=\left[\begin{matrix} 1 & 0 \\ i & 1 \end{matrix}\right],i=\sqrt{-1}$, and $Q=A^{T}BA$,…

Let$B=\left[\begin{matrix} 1 & 3 & \alpha\, \\ 1 & 2 & 3 \\ \alpha\, & \alpha\, & 4 \end{matrix}\right],\alpha\,>2$be the adjoint of a matrix$A$and$\left|A\right|=2.$Then$\left[\begin{matrix} \alpha\, & -2\alpha\, &…