Question

Find the inverse of the matrix:
$open square brackets table row 2 cell space space space space space 1 end cell cell space space space space space 3 end cell row 4 cell space minus 1 end cell cell space space space space space 0 end cell row cell negative 7 end cell cell space space space space space 2 end cell cell space space space space 1 end cell end table close square brackets$

Solution

straight A space equals space open square brackets table row cell space space 2 end cell cell space space space 1 end cell cell space space space space space 3 end cell row cell space space 4 end cell cell space space minus 1 end cell cell space space space space space 0 end cell row cell negative 7 end cell cell space space space space 2 end cell cell space space space space space 1 end cell end table close square brackets

therefore space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row cell space space 2 end cell cell space space 1 end cell cell space space space space 3 end cell row cell space space space 4 end cell cell space minus 1 end cell cell space space space space 0 end cell row cell negative 7 end cell cell space space space space 2 end cell cell space space space space 1 end cell end table close vertical bar space equals space 2 open vertical bar table row cell negative 1 end cell cell space space space 0 end cell row 2 cell space space space 1 end cell end table close vertical bar minus 1 open vertical bar table row 4 cell space space 0 end cell row cell negative 7 end cell cell space space 1 end cell end table close vertical bar plus 3 open vertical bar table row 4 cell space space space minus 1 end cell row cell negative 7 end cell cell space space space space space 2 end cell end table close vertical bar space space space space space space space space space space space space space space equals 2 left parenthesis negative 1 minus 0 right parenthesis space minus 1 left parenthesis 4 minus 0 right parenthesis plus 3 left parenthesis 8 minus 7 right parenthesis space equals negative 2 minus 4 plus 3 space equals space minus 3 not equal to 0.

∴ A^–1 exists
Co-factors of the elements of the first row of | A | are

open vertical bar table row cell negative 1 end cell cell space space space 0 end cell row 2 cell space space space 1 end cell end table close vertical bar comma space space minus open vertical bar table row 4 cell space space space space 0 end cell row cell negative 7 end cell cell space space space 1 end cell end table close vertical bar comma space space space open vertical bar table row cell negative 4 end cell cell space space space minus 1 end cell row cell negative 7 end cell cell space space space space space space 2 end cell end table close vertical bar

i.e. – 1, – 4, 1 respectively.
Co-factors of the elements of the second row of | A | are

negative open vertical bar table row 1 cell space space space 3 end cell row 2 cell space space space 1 end cell end table close vertical bar comma space space space open vertical bar table row 2 cell space space space space space 3 end cell row cell negative 7 end cell cell space space space space 1 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space space space space 1 end cell row cell negative 7 end cell cell space space space space space 2 end cell end table close vertical bar straight i. straight e. space space space 5 comma space space 23 comma space space minus 11 space respectively. Co minus factors space of space the space elements space of space the space third space row space of space vertical line space straight A space vertical line space are open vertical bar table row 1 cell space space space 3 end cell row cell negative 1 end cell cell space space space 0 end cell end table close vertical bar comma space space space space open vertical bar table row 2 cell space space space 3 end cell row 4 cell space space 0 end cell end table close vertical bar space open vertical bar table row 2 cell space space space space space space 1 end cell row 4 cell space space minus 1 end cell end table close vertical bar straight i. straight e. space 3 comma space space space 12 comma space – space 6 space space space respectively adj. space straight A space equals space open square brackets table row cell negative 1 end cell cell space space space minus 4 end cell cell space space space space space space space space 1 end cell row 5 cell space space space 23 end cell cell space space minus 11 end cell row 3 cell space space 12 end cell cell space space space space minus 6 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 1 end cell cell space space 5 end cell cell space space space space 3 end cell row cell negative 4 end cell cell space space 23 end cell cell space space space space 12 end cell row 1 cell space minus 11 end cell cell space space minus 6 end cell end table close square brackets Now 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 minus 1 third open square brackets table row cell negative 1 end cell cell space space space space 5 end cell cell space space space space 3 end cell row cell negative 4 end cell cell space space space space 23 end cell cell space space space 12 end cell row 1 cell space space minus 11 end cell cell space space minus 6 end cell end table close square brackets.

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