A real matrix P of size 3 × 3 satisfies P² = P and has trace 2. Compute det(P + 2I).

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A matrix that repeats itself

Linear algebraHard

A real matrix $P$ of size $3 \times 3$ satisfies $P^{2} = P$ and has trace $2$ . Compute $det (P + 2 I)$ .

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Hint

First find which values an eigenvalue of $P$ can take, then let the trace count them.

Solution

If $P v = λ v$ with $v \neq = 0$ , then $P^{2} v = λ^{2} v$ must equal $P v = λ v$ , so $λ^{2} = λ$ and every eigenvalue is $0$ or $1$ .
The trace is the sum of the eigenvalues counted with multiplicity, so exactly $2$ of them equal $1$ and the remaining one equals $0$ .
The eigenvalues of $P + 2 I$ are then $3$ , $3$ and $2$ .
Its determinant is their product, $3 \times 3 \times 2 = 18$ .

Answer: $18$