Exercise · ZW-DIAGONALIZATION-004 · ★☆☆ Basic
A diagonalizable matrix \(A=PDP^{-1}\) has \(D=\operatorname{diag}(2,-1)\). What is the diagonal matrix of \(A^3\) in the same basis?
Choose one option.
Show answer
- B. \(\operatorname{diag}(8,-1)\)
Explanation
\(A^3=PD^3P^{-1}\) and \(D^3=\operatorname{diag}(2^3,(-1)^3)=\operatorname{diag}(8,-1)\).
Review the concept: Diagonalization
A square matrix \(A\) is diagonalizable if it is similar to a diagonal matrix \(D\), that is, if it can be written as an invertible matrix \(P\) times \(D\) times the inverse of \(P\).
\[A=PDP^{-1}\]
More practice problems: diagonalization
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