Matrices

$A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{array}\right] \Rightarrow A^2-2 A=$

Assertion (A): If $B$ is a $3 \times 3$ matrix and $|B|=6$, then $|\operatorname{Adj}(B)|=36$ Reason (R): If $B$ is a square matrix of order $n$, then $|\operatorname{Adj}(B)|=|B|^{\mathbf{n}}$

Let $A=\left[\begin{array}{cccc}1 & 0 & -1 & -3 \\ 0 & 1 & 1 & k-1 \\ 0 & 0 & k-1 & 1\end{array}\right]$ and $k \in R$. Then, the value of $k$, if exists, for which the rank of $A$ is 2 , is

If $\mathrm{A}$ and $\mathrm{B}$ are non-singular matrices and $\operatorname{det}(\mathrm{AB})=(\operatorname{det} \mathrm{A})$ $(\operatorname{det} \mathrm{B})$, then $((\operatorname{det}…

If $A=\left[\begin{array}{cc}\sin \alpha & -\cos \alpha \\ \cos \alpha & \sin \alpha\end{array}\right]$ and $A+A^{-1}=I$, then $\alpha=$

If $A=\left[\begin{array}{ccc}1 & -3 & 2 \\ -2 & 1 & 3 \\ 3 & 2 & -1\end{array}\right]$ then $A^2 \operatorname{Adj} A=$

If the rank of the matrix $A=\left[\begin{array}{cccc}1 & 2 & 1 & -1 \\ -1 & 2 & 3 & 5 \\ 0 & 1 & k & k\end{array}\right]$is 2 and $\mathrm{k}$ is a real number, then $\mathrm{k}$ is a root of the following quadratic…

A matrix whose elements $a_{i j}$ are defined by $a_{i j}=\frac{1}{3}|i-5 j|, i, j=1,2,3 \text { is }$

The rank of the matrix $A=\left[\begin{array}{cccc}1 & 1 & 1 & 3 \\ 2 & 2 & -1 & 3 \\ 1 & 1 & -1 & 1\end{array}\right]$ is

Consider two systems of 3 linear equations in 3 unknowns $\mathrm{AX}=\mathrm{B}$ and $C X=D$. If $A X=B$ has unique solution $D$ and $C X=D$ has unique solution $B$, then the solution of…

If $\alpha, \beta, \gamma(\alpha \lt \beta \lt \gamma)$ are the values of $x$ such that $\left[\begin{array}{ccc}x-2 & 0 & 1 \\ 1 & x+3 & 2 \\ 2 & 0 & 2 x-1\end{array}\right]$ is a singular matrix then $2 \alpha+3…

If $A=\left[\begin{array}{ll}x & 0 \\ 1 & 1\end{array}\right]$ and $B=\left[\begin{array}{ll}8 & 0 \\ 7 & 1\end{array}\right]$ and $A^3=B$, then $x=$