If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^–1) = (det A)^k.

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Solution

straight A to the power of negative 1 end exponent space equals space fraction numerator Adj space straight A over denominator vertical line straight A vertical line end fraction therefore space vertical line straight A to the power of negative 1 end exponent vertical line space equals space fraction numerator vertical line Adj space straight A vertical line over denominator vertical line straight A vertical line end fraction equals space fraction numerator vertical line straight A vertical line to the power of 3 minus 1 end exponent over denominator vertical line straight A vertical line end fraction left square bracket because space If space straight A space is space straight a space non space singular space matrix space of space order space straight n comma space then space vertical line adj space left parenthesis straight A right parenthesis space equals space vertical line straight A vertical line to the power of straight n minus 1 end exponent right square bracket equals space fraction numerator vertical line straight A vertical line squared over denominator vertical line straight A vertical line end fraction As space we space are space given space that space vertical line straight A to the power of negative 1 end exponent vertical line space equals space vertical line straight A vertical line to the power of straight k therefore space straight k space equals negative 1

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