Let $B$ and $C$ be $n \times n$ matrices such that $A=B+C$, $BC=CB$, and $C^2=0$ (where $0$ is the null matrix). Then, $B^{2020}[B+(2021)C]=$

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]$,$A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=PAP^{T}$. If $P^{T}Q^{2007}P=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$,then $2a+b-3c-4d$ is equal to:

If $A$ is a square matrix of order $3$ and $A^2+A+2I=0$,then

If $A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}$, then the determinant of the matrix $(A^{2025} - 3A^{2024} + A^{2023})$ is

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1} = \operatorname{adj}(\operatorname{adj} M)$,then which of the following statement$(s)$ is/are $ALWAYS \text{ } TRUE$?

Let $\alpha \beta \gamma = 45$; $\alpha, \beta, \gamma \in R$. If $x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0)$ for some $x, y, z \in R$ such that $xyz \neq 0$,then $6\alpha + 4\beta + \gamma$ is equal to: