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Adjoint and Inverse of a Matrix — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAdjoint and Inverse of a Matrix1 Mark
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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