If $A = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix}$,then find the value of $|A B B'|$.

For $M=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and for any $n \in N$,the matrix $M^{n+1}-M^n=$

Let $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{bmatrix}$. Then $A^{2025}-A^{2020}$ is equal to:

If $A$ and $B$ are both $3 \times 3$ matrices,then which of the following statements are true?
$(i)$ $AB=0 \Rightarrow A=0$ or $B=0$
(ii) $AB=I_3 \Rightarrow A^{-1}=B$
(iii) $(A-B)^2=A^2-2AB+B^2$

If $a, b, c$ are three complex numbers such that $a^2 + b^2 + c^2 = 0$ and $\begin{vmatrix} (b^2 + c^2) & ab & ac \\ ab & (c^2 + a^2) & bc \\ ac & bc & (a^2 + b^2) \end{vmatrix} = K a^2 b^2 c^2$,then the value of $K$ is:

Let $A = \begin{bmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{bmatrix}$ and $B = A - I$. If $\omega = \frac{\sqrt{3}i - 1}{2}$, then the number of elements in the set $\{n \in \{1, 2, \ldots, 100\} : A^n + (\omega B)^n = A + B\}$ is equal to $..........$