Q: If is an idempotent matrix, prove that the determinant of the matrix is either or .
A: Suppose is idempotent. Then, by definition ; thus, has either a column of zeros, a row of zeros, or is the identity matrix (we will omit the trivial case when is the zero matrix).
Case 1: If has a row of zeros
Case 2: If has a column of zeros
Case 3: If is the identity matrix, then each of the determinant is equal to or . Clearly, when or . Conversely, only if and . Observe that, only if is the minor of some along the diagonal. Moreover, only appears when (along the diagonal); thus, will always be positive. Therefore, regardless of whether you expand by a row or column, and only when which occurs exactly once in any row or column; as a result, for any .
Does that work? I know you can multiply all the terms of the diagonal to find the determinant if you have an upper or lower traingular, so it makes sense that if . Even so, I am not sure if I described everything correctly. Also, I am not sure I covered every case, given .