Order 6 of the following sentences so that they form a proof by CONTRAPOSITIVE of the statement: 𝐴 is a square matrix. Prove that if 𝐴^2=𝐴, then either 𝐴=𝐼 or 𝐴 is singular.

Question

Order 6 of the following sentences so that they form a proof by CONTRAPOSITIVE 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 that A is invertible and A≠I.
3. If it were the case, then since A is invertible we could multiply both sides by A^-1.
4. That would mean that A^-1A²=A^-1A=I
5. Could it be the case that A²=A?
6. But we know that A≠I, so we also know that A²≠A
7. Assume A=I or A is nonsingular
8. Then A must be equal to A²
9. Assume A≠I or A is singular
10. Which would mean that A=I

Answer 1

The contrapositive statement would be: If A is invertible and A is not the identity matrix, then A²≠A. We will proceed through a contradiction proof via step 5:

2, 5, 3, 4, 10, 6