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Choose the correct option.
If A = (a 0 0; 0 a 0; 0 0 a), then Aⁿ equals to (Understanding)
If $A = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix}$, then $A^n$ equals to (Understanding)
- (a)$\begin{pmatrix} a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n \end{pmatrix}$
- (b)$\begin{pmatrix} a & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a \end{pmatrix}$
- (c)$\begin{pmatrix} a^n & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a^n \end{pmatrix}$
- (d)$\begin{pmatrix} na & 0 & 0 \\ 0 & na & 0 \\ 0 & 0 & a^n \end{pmatrix}$
Answer
Answer
Official answer keyCorrect option: (a)
Answer: (a) $\begin{pmatrix} a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n \end{pmatrix}$
$A = aI$, so $A^n = a^n I^n = a^n I$.