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

Solution

straight A space equals space open square brackets table row 2 cell space space space space space space space 3 end cell row 1 cell space space minus 4 end cell end table close square brackets comma space space space straight B space equals space open square brackets table row cell space space 1 end cell cell space space minus 2 end cell row cell negative 1 end cell cell space space space space space 3 end cell end table close square brackets open vertical bar straight A close vertical bar space equals space open vertical bar table row 2 cell space space space 3 end cell row 1 cell space space minus 4 end cell end table close vertical bar space equals space minus 8 minus 3 space equals space minus 11 space not equal to space 0 straight A subscript 11 space equals space minus 4 comma space space straight A subscript 12 equals negative 1 comma space space straight A subscript 21 space equals space minus 3 comma space space straight A subscript 22 space equals space 2 adj. space straight A space equals space open square brackets table row cell straight A subscript 11 end cell cell space space space straight A subscript 12 end cell row cell straight A subscript 21 end cell cell space space space straight A subscript 22 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 4 end cell cell space space space space minus 1 end cell row cell negative 3 end cell cell space space space space space space 2 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 4 end cell cell space space space space minus 3 end cell row cell negative 1 end cell cell space space space space space space space 2 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 space equals space minus 1 over 11 open square brackets table row cell negative 4 end cell cell space space space space minus 3 end cell row cell negative 1 end cell cell space space space space space space space 2 end cell end table close square brackets open vertical bar straight B close vertical bar space equals space open vertical bar table row 1 cell space space space minus 2 end cell row cell negative 1 end cell cell space space space space space 3 end cell end table close vertical bar space equals space 3 minus 2 space equals space 1 space not equal to space space 0 space space space space straight B subscript 11 space equals space 3 comma space space straight B subscript 12 space equals space 1 comma space space space space space straight B subscript 21 space equals space 2 comma space space space straight B subscript 22 space equals space 1 adj. space straight B space equals space open square brackets table row cell straight B subscript 11 end cell cell space 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 space equals space open square brackets table row 3 cell space space 1 end cell row 2 cell space space space 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 3 cell space space space 2 end cell row 1 cell space space space 1 end cell end table close square brackets 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 space open square brackets table row 3 cell space space space 2 end cell row 1 cell space space 1 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 over 11 open square brackets table row 3 cell space space space space 2 end cell row 1 cell space space space space 1 end cell end table close square brackets space space space space open square brackets table row cell negative 4 end cell cell space space space space minus 3 end cell row cell negative 1 end cell cell space space space space space space 2 end cell end table close square brackets space equals space minus 1 over 11 open square brackets table row cell negative 12 minus 2 end cell cell space space space minus 9 plus 4 end cell row cell negative 4 minus 1 end cell cell negative 3 plus 2 end cell end table close square brackets

therefore 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 over 11 open square brackets table row cell negative 14 end cell cell space space space minus 5 end cell row cell negative 5 end cell cell space space space minus 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 space space space space space space space... left parenthesis 1 right parenthesis space space space space space space space space space space space space space space space space space space AB space equals space open square brackets table row 2 cell space space space space space space 3 end cell row 1 cell space space minus 4 end cell end table close square brackets space space open square brackets table row 1 cell space space space minus 2 end cell row cell negative 1 end cell cell space space space space space 3 end cell end table close square brackets space equals space open square brackets table row cell 2 minus 3 end cell cell space space space space minus 4 plus 9 end cell row cell 1 plus 4 end cell cell space space space minus 2 minus 12 end cell end table close square brackets space equals space open square brackets table row cell negative 1 end cell cell space space 5 end cell row 5 cell space space minus 14 end cell end table close square brackets space space space space space space space space space space space space space space space open vertical bar AB close vertical bar space equals space open vertical bar table row cell negative 1 end cell cell space space space space 5 end cell row 5 cell negative 14 end cell end table close vertical bar space equals space 14 minus 25 space equals space minus 11 space not equal to space 0

Co-factors of elements of first row of | AB | and –14, –5 respectively.
Co-factors of elements ofrsecond row of | AB | and –5, –1 respectively.
$space space therefore space space space space adj. space left parenthesis AB right parenthesis space equals space open square brackets table row cell negative 14 end cell cell space space space space minus 5 end cell row cell negative 5 end cell cell space space space space minus 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 14 end cell cell space space space minus 5 end cell row cell negative 5 end cell cell space space minus 1 end cell end table close square brackets space space space space space space space space space space left parenthesis AB right parenthesis to the power of negative 1 end exponent space equals fraction numerator adj. space left parenthesis AB right parenthesis over denominator open vertical bar AB close vertical bar end fraction space equals space minus 1 over 11 open square brackets table row cell negative 14 end cell cell space space minus 5 end cell row cell negative 5 end cell cell negative 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis$
From (1) and (2), we get, (AB)^–1 = B^–1 A^–1

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