If A and B are square matrices such that B = -A^ -1 BA, then (A +… - Vidyadip

MCQ

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

A
$O$
✓
$A^2 + B^2$
C
$A^2 + 2AB + B^2$
D
$A + B$

✓

Answer

Correct option: B.

$A^2 + B^2$

$B = -A^{-1} BA$
$\Rightarrow AB = -AA^{-1}BA$
$\Rightarrow Ab = -IBA$
$\Rightarrow AB = -BA$
$\Rightarrow AB + BA = 0 .....(i)$
Consider,
$(A + B)^2 = A^2 + AB + BA + B^2$
$(\because\text{AB}\neq\text{BA})$
$(A + B)^2 = A^2 + O + B^2$
$(A + B)^2 = A^2 + B^2$

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