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3 and 4 .Determinants and Matrices
normal
If $A$ is matrix such that $A^2 + A + 2I = O$, then which of the following is $INCORRECT$ ?
A
$A$ is non-singular
B
$A \neq O$
C
$A$ is symmetric
D
$A^{-1} = -(A + I)$ (Where $I$ is unit matrix of order $2$ and $O$ is null matrix of order $2$ )
Solution
We have $A (A + I) = – 2I$
==>$|A (A + I) | = | – 2I |$
==>$| A | | A + I | = 20$
Thus ,$ | A | \neq 0$
Also , $A$ $\left\{ { – \,\,\frac{1}{2}\,(A + I)} \right\}\, = \,I\,$ $= I$
$==>A^{-1} =$ $ – \,\,\frac{1}{2}\,\,(A + I)\,$ Clearly $A \neq O$ for otherwise $| A | = 0$
Standard 12
Mathematics
