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

We have,

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

$⇒ AB=-A (A^{-1} BA)$

$⇒ AB=-((AA^{-1}) (BA))$

$⇒ AB=-(I (BA))$

$⇒ AB=-BA$

$⇒AB+ BA=0$

Now,

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

$⇒(A + B)^2 =A^2 + AB + BA + B^2$

$⇒(A + B)^2 =A^2 +O+ B^2$

$⇒(A + B)^2 =A^2+ B^2$