Concept:We are given that A is a square matrix with the property A² = A, known as an idempotent matrix. This property allows us to simplify all higher powers of A to A itself. The goal is to evaluate the expression A(I − 2A)³ + 2A³.
Explanation:Step 1: Simplify powers of A. Since A² = A, then A³ = A·A² = A·A = A. Similarly, any power A^n (n ≥ 1) equals A.
Step 2: Expand (I − 2A)³ using the binomial theorem:
(I−2A)3=I3−3⋅I2⋅(2A)+3⋅I⋅(2A)2−(2A)3=I−6A+12A2−8A3.Step 3: Substitute A² = A and A³ = A into the expansion:
(I−2A)3=I−6A+12A−8A=I−2A.Step 4: Multiply by A:
A(I−2A)3=A(I−2A)=A−2A2.Since A² = A, this becomes
A−2A=−A.Step 5: Add 2A³. Because A³ = A, we have
2A3=2A. Hence the full expression is:
−A+2A=A.Answer:The expression simplifies to A. Therefore, the correct option is
D. A.