Question-46

Select all true statements concerning an idempotent matrix \(A\).

• \(A^{n} = A\), for \(n \geq 2\) and \(n \in \mathbb{N}\)
• If \(\lambda\) is an eigenvalue of \(A\), then \(\lambda = 0, 1\)
• If \(A\) is invertible, then \(A = I\)
• \(|\text{det}(A)| > 1\)
• \(A\) is full rank.
• \(A\) is symmetric.

Answer

• \(A^{n} = A\), for \(n \geq 2\) and \(n \in \mathbb{N}\)
• If \(\lambda\) is an eigenvalue of \(A\), then \(\lambda = 0, 1\)
• If \(A\) is invertible, then \(A = I\)
• \(|\text{det}(A)| > 1\)
• \(A\) is full rank.
• \(A\) is symmetric.

Solution

If \(\displaystyle A\) is idempotent, then \(\displaystyle A^{2} =A\).

• It follows by induction that \(\displaystyle A^{n} =A\) for \(\displaystyle n\geqslant 2\) and \(\displaystyle n\in \mathbb{N}\).
• If \(\displaystyle A\) is invertible, we have:

\[\begin{equation*} \begin{aligned} A & =AI\\ & =AAA^{-1}\\ & =A^{2} A^{-1}\\ & =AA^{-1}\\ & =I \end{aligned} \end{equation*}\]

• If \(\displaystyle ( \lambda ,v)\) is an eigenpair of \(\displaystyle A\):

\[\begin{equation*} \begin{aligned} A^{2} v & =Av\\ \lambda ^{2} v & =\lambda v\\ \lambda ( \lambda -1) v & =0 \end{aligned} \Longrightarrow \lambda =0,1 \end{equation*}\]