Let $straight A space equals space open square brackets table row 3 cell space space space 7 end cell row 2 cell space space 5 end cell end table close square brackets space and space straight B space equals space open square brackets table row 6 cell space space 8 end cell row 7 cell space space 9 end cell end table close square brackets comma$ verify that (AB)^–1 = B^–1 A^–1.

Solution

straight A space equals space open square brackets table row 3 cell space space 7 end cell row 2 cell space space 5 end cell end table close square brackets comma space space space straight B equals space open square brackets table row 6 cell space space space 8 end cell row 7 cell space space space 9 end cell end table close square brackets space space space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 3 cell space space 7 end cell row 2 cell space space 5 end cell end table close vertical bar space equals space 15 minus 14 space equals space 1 space not equal to space 0 space space space space space space straight A subscript 11 space equals space 5 comma space space straight A subscript 12 space equals space minus 2 comma space space space straight A subscript 21 equals space minus 7 comma space space straight A subscript 22 space equals space 3 adj. space straight A space equals space open square brackets table row cell straight A subscript 11 end cell cell space space straight A subscript 12 end cell row cell straight A subscript 21 end cell cell straight A subscript 22 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 5 cell space space space minus 2 end cell row cell negative 7 end cell cell space space space space space space 3 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 5 cell space space minus 7 end cell row cell negative 2 end cell cell space space space space 3 end cell end table close square brackets 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 open square brackets table row cell space space space 5 end cell cell space space space minus 7 end cell row cell negative 2 end cell cell space space space space space 3 end cell end table close square brackets space space space space space space straight B space equals space open square brackets table row 6 cell space space space space 8 end cell row 7 cell space space space 9 end cell end table close square brackets open vertical bar straight B close vertical bar space equals space open vertical bar table row 6 cell space space space 8 end cell row 7 cell space space 9 end cell end table close vertical bar space equals space 54 minus 56 space equals space minus 2 space not equal to 0 space space space straight B subscript 11 space equals 9 comma space space straight B subscript 12 space equals space minus 7 comma space space straight B subscript 21 space equals space minus 8 comma space space space straight B subscript 22 space equals space 6 space space adj. space straight B space equals space open square brackets table row cell straight B subscript 11 end cell cell space straight B subscript 12 end cell row cell straight B subscript 21 end cell cell straight B subscript 22 end cell end table close square brackets to the power of apostrophe space equals open square brackets table row 9 cell space space space space minus 7 end cell row cell negative 8 end cell cell space space space space space space 6 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 9 cell space space space space minus 8 end cell row cell negative 7 end cell cell space space space space space space 6 end cell end table close square brackets 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 space minus 1 half open square brackets table row 9 cell space space space minus 8 end cell row cell negative 7 end cell cell space space space space space 6 end cell end table close square brackets therefore space 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 half open square brackets table row 9 cell space space space space minus 8 end cell row cell negative 7 end cell cell space space space space space space 6 end cell end table close square brackets space open square brackets table row 5 cell space space minus 7 end cell row cell negative 2 end cell cell space space space space 3 end cell end table close square brackets space equals space minus 1 half open square brackets table row cell 45 plus 16 end cell cell space space minus 63 minus 24 end cell row cell negative 35 minus 12 end cell cell space space 49 plus 18 end cell end table close square brackets

therefore 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 equals space minus 1 half open square brackets table row 61 cell space space space space minus 87 end cell row cell negative 47 end cell cell space space space space 67 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 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 AB space equals space open square brackets table row 3 cell space 7 end cell row 2 cell space 5 end cell end table close square brackets space open square brackets table row 6 cell space 8 end cell row 7 cell space 9 end cell end table close square brackets space equals space open square brackets table row cell 18 plus 49 end cell cell space space 24 plus 63 end cell row cell 12 plus 35 end cell cell space space 16 plus 45 end cell end table close square brackets space equals space open square brackets table row 67 cell space space space 87 end cell row 47 cell space space 61 end cell end table close square brackets open vertical bar AB close vertical bar space equals space open vertical bar table row 67 cell space space 87 end cell row 47 cell space 61 end cell end table close vertical bar space equals space 4087 minus 4089 space equals space minus 2 not equal to 0

Co-factors of the elements of first row of | AB j and 61, –47 respectively.
Co-factors of the elements of second row of | AB | and –87, 67 respectively.
$therefore space space adj. space left parenthesis AB right parenthesis space equals space open square brackets table row 61 cell space space space minus 47 end cell row cell negative 87 end cell cell space space space space 67 end cell end table close square brackets to the power of apostrophe space open square brackets table row 61 cell space space minus 87 end cell row cell negative 47 end cell cell space space space space space 67 end cell end table close square brackets therefore space space space space left parenthesis AB right parenthesis to the power of negative 1 end exponent space equals space fraction numerator adj. space left parenthesis AB right parenthesis over denominator open vertical bar AB close vertical bar end fraction equals negative 1 half open square brackets table row 61 cell space space minus 87 end cell row cell negative 47 end cell cell space space space space space 67 end cell end table close square brackets space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis$
From (1) and (2), we get,
(AB)^–1 = B^–1 A^–1.

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