The mth power of a diagonal matrix D a11 0 p 0 0 a22 p 0 ( ( 0 0 p ann is Dm am 11 0 p 0
Chapter 8, Problem 38(choose chapter or problem)
The mth power of a diagonal matrix
\(\mathbf{D}=\left(\begin{array}{cccc} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & \cdots & a_{n n} \end{array}\right) \)
is
\(\mathbf{D}^{m}=\left(\begin{array}{cccc} a_{11}^{m} & 0 & \cdots & 0 \\ 0 & a_{22}^{m} & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & \cdots & a_{n n}^{m} \end{array}\right) \)
Use this result to compute
\(\left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 5 \end{array}\right)^{4} \).
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