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## Homework Statement

Prove that if a matrix A is idempotent (A^2 = A), then the determinant of A is 0 (det(A) = 0).

## Homework Equations

None

## The Attempt at a Solution

1) A^2 = A

2) AA = A

3) AA - A = A - A <- Here, I subtracted A from both sides

4) AA - A = 0 <- property of the Zero matrix (A+(-A) = 0)

5) A(I-A) = 0 <- I factored out A here, not sure if this is legal or not in matrices.

A(I-A) = 0 implies that either I-A = 0 or A = 0; det(0) = 0, therefore, det(A) = 0.