Find (AB)^–1 if $straight A space equals space open square brackets table row 5 cell space space 0 end cell row 2 cell space space space 3 end cell end table close square brackets comma space space space straight B to the power of negative 1 end exponent space equals space open square brackets table row 1 cell space space space 2 end cell row 1 cell space space 4 end cell end table close square brackets$

Solution

Here $straight A space equals open square brackets table row 5 cell space 0 end cell row 2 cell space 3 end cell end table close square brackets comma space space straight B to the power of negative 1 end exponent space equals space open square brackets table row 1 cell space space 2 end cell row 1 cell space space 4 end cell end table close square brackets space space$
$open vertical bar straight A close vertical bar space equals space open vertical bar table row 5 cell space space 0 end cell row 2 cell space space 3 end cell end table close vertical bar space equals space 15 minus 0 space equals space 15 rightwards double arrow space space space space straight A to the power of negative 1 end exponent space exists. space$
Co-factors of elements of first row of | A | are 3, – 2 respectively.
Co-factors of elements of second row of | A | are 0, 5 respectively.
$therefore space space space space adj. space straight A space equals space open square brackets table row 3 cell space space minus 2 end cell row 0 cell space space space space 5 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell space space 3 end cell cell space space space 0 end cell row cell negative 2 end cell cell space space space 5 end cell end table close square brackets therefore space space space 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 over 15 open square brackets table row cell space space 3 end cell cell space space space 0 end cell row cell negative 2 end cell cell space space space 5 end cell end table close square brackets space space space space space left parenthesis AB right parenthesis to the power of negative 1 end exponent space equals space straight B to the power of negative 1 end exponent straight A to the power of negative 1 end exponent space equals space 1 over 15 open square brackets table row 1 cell space space 2 end cell row 1 cell space space 4 end cell end table close square brackets space open square brackets table row cell space space space 3 end cell cell space space space 0 end cell row cell negative 2 end cell cell space space space 5 end cell end table close square brackets space equals space 1 over 15 open square brackets table row cell 3 minus 4 end cell cell space space space 0 plus 10 end cell row cell 3 minus 8 end cell cell space space space space 0 plus 20 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 equals space 1 over 15 open square brackets table row cell negative 1 end cell cell space space space 10 end cell row cell negative 5 end cell cell space space 20 end cell end table close square brackets$

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