If $straight A space equals space open square brackets table row 1 cell space space space space 3 end cell row 2 cell space space space space 7 end cell end table close square brackets space space and space straight B space equals space open square brackets table row 3 cell space space space space 4 end cell row 6 cell space space space space 2 end cell end table close square brackets comma$
verify (AB)^–1 = B ^–1 A^–1

Solution

straight A space equals space open square brackets table row 1 cell space space 3 end cell row 2 cell space space 7 end cell end table close square brackets comma space space straight B space equals space open square brackets table row 3 cell space space space 4 end cell row 6 cell space space space 2 end cell end table close square brackets open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space space 3 end cell row 2 cell space space 7 end cell end table close vertical bar space equals space 7 minus 6 space equals space 1

Co-factors of elements of first row of |A| are 7 and – 2 respectively
Co-factors of elements of second row of |A| are – 3 and 1 respectively
$therefore space space adj. space straight A space equals space open square brackets table row 7 cell space space space minus 2 end cell row cell negative 3 end cell cell space space space space space 1 end cell end table close square brackets to the power of apostrophe space space equals space open square brackets table row cell space 7 end cell cell space space space minus 3 end cell row cell negative 2 end cell cell space space space space space 1 end cell end table close square brackets space space space space space space straight A to the power of negative 1 end exponent space equals space fraction numerator adj space straight A over denominator open vertical bar straight A close vertical bar end fraction equals space open square brackets table row cell space space space space 7 end cell cell space space space space minus 3 end cell row cell negative 2 end cell cell space space space space space 1 end cell end table close square brackets space space space space space space space space open vertical bar straight B close vertical bar space equals space open vertical bar table row 3 cell space space space space 4 end cell row 6 cell space space space space 2 end cell end table close vertical bar space equals space 6 minus 24 space equals space minus 18$
Co-factors of elements of first row of | B | are 2 and – 6 respectively.
Co-factors of elements of second row of | B | are – 4 and 3 respectively.
$therefore space space space space space space adj. space straight B space equals space open square brackets table row cell space space space 2 end cell cell space space space space minus 6 end cell row cell negative 4 end cell cell space space space space space 3 end cell end table close square brackets to the power of apostrophe space space equals space open square brackets table row 2 cell space space space minus 4 end cell row cell negative 6 end cell cell space space space space space 3 end cell end table close square brackets space space space space space space space space space space space space space straight B to the power of negative 1 end exponent space equals space fraction numerator adj space straight B over denominator open vertical bar straight B close vertical bar end fraction space equals negative 1 over 18 open square brackets table row 2 cell space space space minus 4 end cell row cell negative 6 end cell cell space space space space space 3 end cell end table close square brackets therefore space space space thin space straight B to the power of negative 1 end exponent straight A to the power of negative 1 end exponent space equals space minus 1 over 18 open square brackets table row cell space space 2 end cell cell space space space space space minus 4 end cell row cell negative 6 end cell cell space space space space space space space 3 end cell end table close square brackets space space open square brackets table row 7 cell space space space space space minus 3 end cell row cell negative 2 end cell cell space space space space space 1 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 equals negative 1 over 18 space open square brackets table row cell 14 plus 8 end cell cell space space space space minus 6 minus 4 end cell row cell negative 42 minus 6 end cell cell space space space space space space 18 plus 3 end cell end table close square brackets space equals space minus 1 over 18 space open square brackets table row 22 cell space space space minus 10 end cell row cell negative 48 end cell cell space space space space space 21 end cell end table close square brackets therefore space space space space space straight B to the power of negative 1 end exponent straight A to the power of negative 1 end exponent space space equals space minus 1 over 18 open square brackets table row 22 cell space space minus 10 end cell row cell negative 48 end cell cell space space space space space 21 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 space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space space space space space space space space space AB equals space open square brackets table row 1 cell space space space 3 end cell row 2 cell space space space 7 end cell end table close square brackets space space open square brackets table row 3 cell space space space 4 end cell row 6 cell space space space 2 end cell end table close square brackets space equals space open square brackets table row cell 3 plus 18 end cell cell space space space space space space 4 plus 6 end cell row cell 6 plus 42 end cell cell space space space space space space 8 plus 14 end cell end table close square brackets space equals space open square brackets table row 21 cell space space space 10 end cell row 48 cell space space space 22 end cell end table close square brackets open vertical bar AB close vertical bar space equals space space open vertical bar table row 21 cell space space space space space space 10 end cell row 48 cell space space space space space 22 end cell end table close vertical bar space equals space 462 minus 480 space equals negative 18$

Co-factors of elements of first row of | AB | are 22 and – 48 respectively.
Co-factors of elements of second row of | AB | are – 10 and 21 respectively.
$therefore space space adj. space left parenthesis AB right parenthesis space equals space open square brackets table row 22 cell space space space minus 48 end cell row cell negative 10 end cell cell space space space space space 21 end cell end table close square brackets to the power of comma space equals space open square brackets table row 22 cell space space space minus 10 end cell row cell negative 48 end cell cell space space space space space space space 21 end cell end table close square brackets space space space space space space thin space left parenthesis AB right parenthesis to the power of negative 1 end exponent space equals space fraction numerator adj. space left parenthesis AB right parenthesis over denominator open vertical bar AB close vertical bar end fraction space equals space minus 1 over 18 open square brackets table row 22 cell space space space space minus 10 end cell row cell negative 48 end cell cell space space space space space space 21 end cell end table close square brackets space space space space space space space space space space space space... left parenthesis 2 right parenthesis$
From (1) and (2), we get, (AB) ^–1 = B ^–1 A^–1

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