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# True or False: $|\;A^{-1}\;|\neq |\;A\;|^{-1}$,where A is non-singular matrix.

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Toolbox:
• (i) A square matrix A is invertible if and only if A is non-singular matrix.
• (ii) A matrix is said to be non-singular if $|A|\neq 0$
• (iii) $A^{-1}=\frac{1}{|A|}(adj A)$
We know $AA^{-1}=I=A^{-1}A$
Taking determinants on both sides,
$|AA^{-1}|=|I|$
But we know |AB|=|A||B|
Applying this we get,
$|A||A^{-1}|=|I|=1$
Therefore $|A^{-1}|=\frac{1}{|A|}$
$\qquad\qquad\quad\;\;=|A|^{-1}$
Therefore $|A^{-1}|=|A|^{-1}$
So the given statement is False.
answered Mar 27, 2013