Question

If \(A\) and \(B\) are \(n\times n\) square matrices such that \[ (2A+B)^2+(A-3B)^2=5A^2-2AB+10B^2, \] then \(ABAB=\)

• A. \[ \frac{1}{2}\Big[(A-B)^2+(A+B)^2\Big] \]
• B. \(4AB\)
• C. \[ \frac{1}{2}\Big[(A+B)^2-(A-B)^2\Big] \]
• D. \(A^2B^2\)

Hint:

Whenever matrix expressions are expanded and compared, check whether the comparison implies \(AB=BA\). Once matrices commute, many algebraic simplifications become possible.

Answer 1

The Correct Option is D

Step 1: Expand both squared matrix expressions.
Compute (2A + B)² = 4A² + 2AB + 2BA + B² and (A - 3B)² = A² - 3AB - 3BA + 9B². Adding these expansions gives 5A² - AB - BA + 10B².

Step 2: Equate with the provided equation.
The problem states that the sum equals 5A² - 2AB + 10B². Comparing, we obtain -AB - BA = -2AB, which simplifies to AB + BA = 2AB. Rearranging yields BA = AB, establishing that A and B commute.

Step 3: Simplify the product ABAB using commutativity.
With AB = BA, we can rewrite ABAB = A(BA)B. Substituting BA = AB transforms this into A(AB)B = A²B².

Step 4: Final conclusion.
Hence ABAB = A²B².