Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be an $n \times n$ identity matrix. Prove that the matrix $I −A$ is an idempotent matrix

Question

A square matrix $A$ is called idempotent if $A^2 = A$.

(a) Suppose $A$ is an $n × n$ idempotent matrix and let $I$ be the $n × n$ identity matrix. Prove that the matrix $I −A$ is an idempotent matrix.

(b) Assume that $A$ is an $n×n$ non zero idempotent matrix. Then determine all integers $k$ such that the matrix $I − kA$ is idempotent.

I need help. I didn't know what to do... All I know is that I need to show that $(I−A)^2=(I−A)$. I don't have any clue on how I should proceed. Thanks in advance

Answer 1

An idempotent matrix $A$ by definition satisfies

$A^2 = A; \tag 1$

from this we immediately have

$(I - A)^2 = I^2 - 2A + A^2 = I - 2A + A = I - A, \tag 2$

that is, $I - A$ is also idempotent.

Now if $I - kA$ is idempotent, then via (1) we find

$I - kA = (I - kA)^2 = I - 2kA + k^2A^2 = I - 2kA + k^2A, \tag 3$

whence

$(k^2 - k)A = 0; \tag 4$

since $A \ne 0$ this forces

$k^2 - k = 0 \Longrightarrow k = 0, 1; \tag 5$

thus the only candidates for $I - kA$ are

$I - kA = I, I - A. \tag 6$ .