If A = a 0
0 b
then
A^n = a^n 0
0 b^n
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Please see attachment its hard to display the question in text. Thanks.
$\displaystyle
A^1=A=\left[\begin{array}{cc}
a&0\\0&b
\end{array}\right]
$
which will give us a base case.
Suppose true for some $\displaystyle k>1$
Then
$\displaystyle
A^{k+1}=AA^k=\left[\begin{array}{cc}a&0\\0&b\end{array}\right]\left[\begin{array}{cc}a^k&0\\0&b^k\end{array}\right]
$
Now do the multiplication abd you should be there.
CB
Have you ever heard about mathematical induction?
n = 1: OK
n -> n+1
$\displaystyle A^{n+1} = A^n \cdot A = \begin{pmatrix} a^n & 0 \\ 0 & b^n \end{pmatrix}\cdot \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} $
$\displaystyle = \begin{pmatrix} a^n \cdot a + 0 & 0 \\ 0 & b^n \cdot b \end{pmatrix}$
$\displaystyle = \begin{pmatrix} a^{n+1} & 0 \\ 0 & b^{n+1} \end{pmatrix}$
Edit: Too slow