If A and B are two square matrices such that B = - A^ - 1 BA, the… - Vidyadip

MCQ

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

A
$0$
✓
${A^2} + {B^2}$
C
${A^2} + 2AB + {B^2}$
D
$A + B$

✓

Answer

Correct option: B.

${A^2} + {B^2}$

b
(b) Given, $B = - {A^{ - 1}}BA$

$\therefore$ $AB = - A{A^{ - 1}}BA = - IBA = - BA$

$\therefore$ $AB = - BA$

Now ${(A + B)^2} = (A + B)(A + B)$

= ${A^2} + AB + BA + {B^2}$

= ${A^2} + {B^2}$ [$\because$ $BA$= $-BA$]

Thus, ${(A + B)^2} = {A^2} + {B^2}.$

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