Question

If \( A \) is a square matrix and \( I \) is an identity matrix such that \( A^2 = A \), then \( A(I - 2A)^3 + 2A^3 \) is equal to:

• A. I+A
• B. I+2A
• C. I−A
• D. A

Hint:

When simplifying matrix expressions, always take into account any given properties of the matrices. In this case, we used \( A^2 = A \) (which means \( A \) is idempotent) to simplify the expression. Also, when expanding expressions like \( (I - 2A)^3 \), be sure to apply the binomial theorem carefully and simplify terms step by step. Always check for matrix properties such as \( I^3 = I \) to help streamline the process.

Answer 1

The Correct Option is D

\( A(I - 2A)^3 + 2A^3 \) (Since \( A^2 = A \))

⇒\( \Rightarrow A(I - 2A)^3 + 2A \)

⇒\( A[I^3 - 3I^2(2A) + 3I(2A)^2 - (2A)^3] + 2A \)

⇒\( A[I^3 - 6I^2A + 12IA^2 - 8A^3] + 2A \)

⇒\( A[I^3 - 6I^2A + 12IA^2 - 8A] + 2A \)   we know that \(( I^3= I)\)

⇒\( A[I - 6IA + 12A - 8A] + 2A \)

⇒\( A[I - 14A + 12A] + 2A \)

⇒\( A[I - 2A] + 2A \)

⇒\( AI - 2A^2 + 2A \)

⇒\( A - 2A + 2A \)

⇒\( A \)

Answer 2

Given expression: 

\( A(I - 2A)^3 + 2A^3 \) (Since \( A^2 = A \))

Step 1: Simplify the expression using \( A^2 = A \):

\[ \Rightarrow A(I - 2A)^3 + 2A \]

Step 2: Expand \( (I - 2A)^3 \):

Use the binomial expansion to expand \( (I - 2A)^3 \): \[ (I - 2A)^3 = I^3 - 3I^2(2A) + 3I(2A)^2 - (2A)^3 \] Substitute the powers of \( I \) and simplify: \[ \Rightarrow A[I^3 - 6I^2A + 12IA^2 - 8A^3] + 2A \]

Step 3: Apply known identities \( I^3 = I \) and \( A^2 = A \):

\[ \Rightarrow A[I - 6IA + 12IA^2 - 8A] + 2A \] Since \( A^2 = A \), we replace \( A^2 \) with \( A \): \[ \Rightarrow A[I - 6IA + 12A - 8A] + 2A \]

Step 4: Simplify further:

\[ \Rightarrow A[I - 14A + 12A] + 2A \] Simplify the terms inside the brackets: \[ \Rightarrow A[I - 2A] + 2A \]

Step 5: Final simplification:

\[ \Rightarrow AI - 2A^2 + 2A \] Since \( A^2 = A \), we replace \( A^2 \) with \( A \): \[ \Rightarrow A - 2A + 2A \] Simplifying the terms: \[ \Rightarrow A \]

Conclusion: The simplified result is \( A \).