Linear Algebra Problem
A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix.
a) Show that I − A is also idempotent.
b) Show that if A is invertible, then A = I.
c) Show that the only possible eigenvalues of A are 0 and 1. (Hint: Suppose x is an
eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.)
1. Because A is idempotent, we can use:
A2 = A
( I − A) 2 = I 2 − 2 IA + A2
( I − A) 2 = I − 2 A + A
( I − A) 2 = I − A
I − A Is idempotent
2. Because A is invertible, A−1 exists:
A2 = A
Multiply both sides by A −1
A−1 ( AA) = A−1 A
( A−1 A) A = A−1 A
IA = I
A= I
A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix.
a) Show that I − A is also idempotent.
b) Show that if A is invertible, then A = I.
c) Show that the only possible eigenvalues of A are 0 and 1. (Hint: Suppose x is an
eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.)
1. Because A is idempotent, we can use:
A2 = A
( I − A) 2 = I 2 − 2 IA + A2
( I − A) 2 = I − 2 A + A
( I − A) 2 = I − A
I − A Is idempotent
2. Because A is invertible, A−1 exists:
A2 = A
Multiply both sides by A −1
A−1 ( AA) = A−1 A
( A−1 A) A = A−1 A
IA = I
A= I
