Question

If $A^2+3I=0$ find $A^{-1}$.
Evaluate: $\sin \{\frac{\pi }{2}-{\sin }^{-1}(-\frac{\sqrt{3}}{2}\left)\right\}$

Answer 1

Given that,

$A^2+3I=0$

By multiplying both sides with $A^{-1}$ we get,

$A^2.A^{-1}+3I.A^{-1}=0.A^{-1}$

$A.(A.A^{-1})+3(I.A^{-1})=0$

$A.I+3A^{-1}=0$

$A+3A^{-1}=0$

$3A^{-1}=0-A$

$\therefore A^{-1}=\frac{1}{3}(-A)$