Question

If A and B are two matrices such that AB=B and BA=A then A² + B² =

• A. AB
• B. A+B
• C. 2BA
• D. 2AB

Hint:

In matrix algebra, when dealing with conditions involving products of matrices, be sure to manipulate and combine the equations carefully. Using properties like associativity and distributivity of matrix multiplication can help simplify expressions, as demonstrated in this problem.

Answer 1

The Correct Option is B

The correct answer is (B) : A+B.

Answer 2

The correct answer is: (B) A + B.

We are given two matrices \( A \) and \( B \) such that:

• AB = B
• BA = A

We are asked to find the value of \( A^2 + B^2 \). To solve this, we will use the given conditions: - From \( AB = B \), multiply both sides on the left by \( A \):

\( A(AB) = A \cdot B \)

This simplifies to:

\( A^2B = AB \)

Since \( AB = B \), this becomes:

\( A^2B = B \)

- From \( BA = A \), multiply both sides on the right by \( B \):

\( (BA)B = A \cdot B \)

This simplifies to:

\( AB^2 = AB \)

Since \( AB = B \), this becomes:

\( AB^2 = B \)

Now, we can add the two equations \( A^2B = B \) and \( AB^2 = B \):

\( A^2B + AB^2 = B + B \)

This simplifies to:

\( A^2 + B^2 = A + B \)

Therefore, the correct answer is (B) A + B.