If A and B are square matrices such that

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Matrices

Question:

If A and B are square matrices such that $B=-A^{-1} BA$, then $(A + B)^2 =$

Options:

$O$

$A^2+ B^2$

$A^2+2AB+ B^2$

$A+B$

Correct Answer:

$A^2+ B^2$

Explanation:

We have,

$B=-A^{-1} BA$

$⇒ AB=-A (A^{-1} BA)$

$⇒ AB=-((AA^{-1}) (BA))$

$⇒ AB=-(I (BA))$

$⇒ AB=-BA$

$⇒AB+ BA=0$

Now,

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

$⇒(A + B)^2 =A^2 + AB + BA + B^2$

$⇒(A + B)^2 =A^2 +O+ B^2$

$⇒(A + B)^2 =A^2+ B^2$