Explanation:

Here's how to break it down:

- You’re told that A² = A. That makes A an idempotent matrix.

- You’re looking for (I + A)³ - 7A.

Let’s calculate (I + A)³:

- Expand using the binomial theorem:

(I + A)³ = I³ + 3I²A + 3IA² + A³

Since I is the identity: I³ = I, I² = I, IA² = A²

Also, because A² = A and A³ = A²·A = (A)·A = A² = A

So plugging all that in:

(I + A)³ = I + 3A + 3A + A

= I + 7A

Now, subtract 7A:

(I + 7A) - 7A = I

Let’s look at your options:

- Option 1, 3I: This says the answer is three times the identity—not correct.

- Option 2, O: O means the zero matrix—not correct.

- Option 3, I: The identity matrix— this is the correct answer.

- Option 4, 2I: Two times the identity—not correct.

Summary:

- The correct answer is Option 3, I (the identity matrix).

- The calculation relies on knowing that A² = A and how to expand (I + A)³.

- The other options don’t fit the math.

If you want a shortcut for future questions: whenever your matrix satisfies A² = A and you see powers of A, always try reducing with that property first.