If $C$ and $D$ are two $n \times n$ non-singular matrices over the set of real numbers $\mathbb{R}$ such that $CD = -DC$, then $n$ is:

$A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix} \Rightarrow A^2-2A=$

If $\alpha , \beta \neq 0$ and $f(n) = \alpha^n + \beta^n$ and $\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1 + f(1) & 1 + f(2) & 1 + f(3) \\ 1 + f(2) & 1 + f(3) & 1 + f(4) \end{vmatrix} = K(1 - \alpha)^2 (1 - \beta)^2 (\alpha - \beta)^2$,then $K = \dots$

If $A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$ is expressed as a sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$,then $P^{T}-Q^{T}=$

Let $S =\{ M = [a_{ij}], a_{ij} \in \{0,1,2\}, 1 \leq i, j \leq 2\}$ be a sample space and $A = \{M \in S : M \text{ is invertible}\}$ be an event. Then $P(A)$ is equal to

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2I) - 4(A-I) = O$,where $I$ and $O$ are the identity and null matrices,respectively. If $A^5 = \alpha A^2 + \beta A + \gamma I$,where $\alpha, \beta$ and $\gamma$ are real constants,then $\alpha + \beta + \gamma$ is equal to: