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Determinants

Question
CBSEENMA12034566
Wired Faculty App

Given straight A space equals space open square brackets table row 2 cell space space space space space minus 3 end cell row cell negative 4 end cell cell space space space space space space space 7 end cell end table close square brackets comma compute A–1 and show that 2A–1 = 9 I – A.

Solution

Here,  straight A space equals space open square brackets table row cell space space space space 2 end cell cell space space space space minus 3 end cell row cell negative 4 end cell cell space space space space space space space space 7 end cell end table close square brackets
therefore space space space space space open vertical bar straight A close vertical bar space equals open vertical bar table row cell space space space 2 end cell cell space space space space minus 3 end cell row cell negative 4 end cell cell space space space space space space 7 end cell end table close vertical bar space equals space 14 minus 12 space equals space 2 space not equal to space 0 space space space space space rightwards double arrow space space straight A to the power of negative 1 end exponent space exists
Co-factors of the elements of first row of | A | are 7, 4 respectively
Co-factors of the elements of second row of | A | are 3, 2 respectively
therefore space space space space adj. space space straight A space equals space open square brackets table row 7 cell space space space space 4 end cell row 3 cell space space space space 2 end cell end table close square brackets to the power of apostrophe space space equals space open square brackets table row 7 cell space space space 3 space end cell row 4 cell space space space 2 end cell end table close square brackets therefore space space space space space straight A to the power of negative 1 end exponent space equals space fraction numerator adj. space straight A over denominator open vertical bar straight A close vertical bar end fraction space equals space 1 half open square brackets table row 7 cell space space space space 3 end cell row 4 cell space space space space 2 end cell end table close square brackets 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 2 straight A to the power of negative 1 end exponent open square brackets table row 7 cell space space space space 3 end cell row 4 cell space space space space space 2 end cell end table close square brackets Also space 9 straight I space minus space straight A space equals space 9 open square brackets table row 1 cell space space space 0 end cell row 0 cell space space space space 1 end cell end table close square brackets space minus space open square brackets table row cell space space space 2 end cell cell space space space space space minus 3 end cell row cell negative 4 end cell cell space space space space space space space 7 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space equals open square brackets table row 9 cell space space space 0 end cell row 0 cell space space space 9 end cell end table close square brackets space minus space open square brackets table row 2 cell space space space space space space space space minus 3 end cell row cell negative 4 end cell cell space space space space space space space space space space space 7 end cell end table close square brackets space equals space open square brackets table row 7 cell space space space space 3 end cell row 4 cell space space space space 2 end cell end table close square brackets therefore space space space space 2 straight A to the power of negative 1 end exponent space equals space 9 straight I space minus straight A.

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