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Determinants

Question
CBSEENMA12034586
Wired Faculty App

If A is square matrix such that A3 = I, prove that A is non-singular and find adj. A and prove that A–1 = A2.

Solution
Here space space space space space space straight A cubed space equals space straight I space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight A. space straight A. space straight A space equals space straight I space space space space space space space space space space space rightwards double arrow space space space open vertical bar straight A. space space straight A. space straight A close vertical bar space equals space open vertical bar straight I close vertical bar rightwards double arrow space space space space open vertical bar straight A close vertical bar space open vertical bar straight A close vertical bar space open vertical bar straight A close vertical bar space equals space 1 space space space space space space space space space rightwards double arrow space space space space space space space open vertical bar straight A close vertical bar space space equals 1 rightwards double arrow space space space space space space open vertical bar straight A close vertical bar space not equal to space 0 space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight A space is space non minus singular therefore space space space space space straight A to the power of negative 1 end exponent space space exists. Now space straight A space left parenthesis adj. space straight A right parenthesis space equals space open vertical bar straight A close vertical bar space space straight I space space space space space rightwards double arrow space space space straight A left parenthesis adj. space straight A right parenthesis space equals space straight I space space space space space rightwards double arrow space space space straight A to the power of negative 1 end exponent straight A left parenthesis adj. space straight A right parenthesis space equals space straight A to the power of negative 1 end exponent space straight I rightwards double arrow space space space space adj. space straight A space space equals space straight A to the power of negative 1 end exponent space Now space straight A cubed space equals space straight I space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight A cubed. space straight A to the power of negative 1 end exponent space equals straight I thin space straight A to the power of negative 1 end exponent space space space rightwards double arrow space space space straight A squared space equals space straight A to the power of negative 1 end exponent space space rightwards double arrow space space straight A to the power of negative 1 end exponent space equals straight A squared space space space space space space space

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