Order 8 of the following sentences so that they form a direct proof of the statement: 𝐴 is a square matrix. Prove that if 𝐴2=𝐴, then either 𝐴=𝐼 or 𝐴 is singular.

Question

Order 8 of the following sentences so that they form a direct proof of the statement:

A is a square matrix. Prove that if , then either  or  is singular.

Options:

1. Let A be an m×n matrix.
2. Assume A²=A.
3. Then A must be the zero matrix.
4. Case 1: Suppose A is invertible.
5. Since A²=A, then A^-1A²=A^-1A and so A=I
6. Case 2: Suppose A is noninvertible
7. Then A is nonsingular
8. Then ƎA^-1 so that AA^-1=A^-1A=I
9. Then A is singular
10. So in either case, either A=I or A is singular
11. Then A is invertible
12. We know that A is either invertible or noninvertible

Answer 1

2, 12, 4, 8, 5, 6, 9, 10