Q.If $A = \begin{bmatrix}0 & 0\\0 & 5\end{bmatrix}$, then $A^{12}$ is :

For a diagonal matrix, any positive integer power is obtained simply by raising each diagonal entry to that power; here the $(1,1)$ entry is $0$ and the $(2,2)$ entry is $5$, so $A^{12}$ has diagonal entries $0^{12}=0$ and $5^{12}$.

1. $A=\begin{bmatrix}0&0\\0&5\end{bmatrix}$ is a diagonal matrix.
2. For a diagonal matrix $D=\text{diag}(d_1,d_2)$, the product $D\cdot D$ is again diagonal: $D^2=\text{diag}(d_1^2,d_2^2)$, and in general $D^n=\text{diag}(d_1^n,d_2^n)$ for any positive integer $n$ (verified by induction: multiplying two diagonal matrices just multiplies corresponding diagonal entries, since all off-diagonal products vanish).
3. Here $d_1=0,\ d_2=5$, and $n=12$. …