1. If A is a square matrix such that A2 = A, then (I – A)3 + A is equal to

(a) I

(b) 0

(c) I – A

(d) I + A

We are given that \( A \) is a square matrix such that \( A^2 = A \), which means \( A \) is idempotent.

We need to compute \( (I - A)^3 + A \), where \( I \) is the identity matrix.

### Step-by-Step Solution:

1. **Expand \( (I - A)^3 \) using the binomial theorem:**

\[
(I - A)^3 = (I - A)(I - A)(I - A)
\]

We can calculate this incrementally:

\[
(I - A)^2 = (I - A)(I - A) = I^2 - 2A + A^2
\]

Since \( A^2 = A \):

\[
(I - A)^2 = I - 2A + A = I - A
\]

Now, multiply by \( (I - A) \) again:

\[
(I - A)^3 = (I - A)(I - A)^2 = (I - A)(I - A) = I - A
\]

Thus:

\[
(I - A)^3 = I - A
\]

2. **Now add \( A \) to the expression:**

\[
(I - A)^3 + A = (I - A) + A
\]

The terms \( -A \) and \( +A \) cancel out:

\[
(I - A) + A = I
\]

Thus, the expression simplifies to \( I \).

### Final Answer:

The value of \( (I - A)^3 + A \) is \( \boxed{I} \).

---

Would you like further details or have any questions?

Here are five related questions to explore this concept further:

1. What are the properties of an idempotent matrix?
2. How does the binomial expansion work for matrix expressions?
3. Can a matrix be both idempotent and invertible?
4. What happens if \( A^2 = A \) but \( A \neq A^2 \)?
5. How does this result change for non-square matrices?

**Tip:** In matrix algebra, idempotent matrices simplify many computations due to their property \( A^2 = A \), which helps reduce higher powers of the matrix to simpler forms.

If A is a square matrix such that A^2 = A, then (I – A)^3 + A is equal to
(a) I
(b) 0
(c) I – A
(d) I + A

Explore how to solve the matrix algebra problem where A is an idempotent matrix (A^2 = A) and (I - A)^3 + A is simplified to I. This step-by-step solution covers key matrix concepts like idempotent matrices, matrix powers, and binomial expansion. Suitable for advanced high school or college students.

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