Question

If $straight A space equals space open square brackets table row 2 cell space space space 2 end cell cell space space space 1 end cell row cell negative 2 end cell cell space space space 1 end cell cell space space 2 end cell row 1 cell space minus 2 end cell cell space space 2 end cell end table close square brackets space space space and space straight B space equals space open square brackets table row 1 cell space space 3 end cell cell space space 2 end cell row 1 cell space space space 1 end cell cell space 1 end cell row 2 cell negative 3 end cell cell space 1 end cell end table close square brackets comma$ 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 2 end cell cell space space space 1 end cell row cell negative 2 end cell cell space space space space 1 end cell cell space space space 2 end cell row 1 cell space space minus 2 end cell cell space space space 2 end cell end table close square brackets comma space space space straight B space equals space open square brackets table row 1 cell space space space space 3 end cell cell space space space space 2 end cell row 1 cell space space space space 1 end cell cell space space space space 1 end cell row 2 cell space minus 3 end cell cell space minus 1 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 space 2 end cell cell space space space space 1 end cell row cell negative 2 end cell cell space space space space 1 end cell cell space space space space 2 end cell row 1 cell space space minus 2 end cell cell space space space space 2 end cell end table close vertical bar space equals space 2 open vertical bar table row 1 cell space space space space 2 end cell row cell negative 2 end cell cell space space space space 2 end cell end table close vertical bar minus 2 open vertical bar table row cell negative 2 end cell cell space space space space 2 end cell row 1 cell space space space space 2 end cell end table close vertical bar space plus space 1 open vertical bar table row cell negative 2 end cell cell space space space space space space 1 end cell row 1 cell space space minus 2 end cell end table close vertical bar space space space space space space space space space equals 2 space left parenthesis 2 plus 4 right parenthesis space minus 2 space left parenthesis negative 4 minus 2 right parenthesis plus 1 space left parenthesis 4 minus 1 right parenthesis space equals 12 plus 12 plus 3 space equals 27 not equal to 0 straight A subscript 11 space equals space open vertical bar table row 1 cell space space space 2 end cell row cell negative 2 end cell cell space space space 2 end cell end table close vertical bar space equals space 2 space plus space 4 space equals 6 straight A subscript 12 space equals space minus open vertical bar table row cell negative 2 end cell cell space space space space 2 end cell row 1 cell space space 2 end cell end table close vertical bar space equals space minus left parenthesis 4 minus 2 right parenthesis space equals space 6 straight A subscript 13 space equals space open vertical bar table row cell negative 2 end cell cell space space space 1 end cell row 1 cell space space minus 2 end cell end table close vertical bar space space equals space 4 minus 1 space equals space 3 straight A subscript 21 space equals negative open vertical bar table row 2 cell space space space 1 end cell row cell negative 2 end cell cell space space 2 end cell end table close vertical bar space equals space minus left parenthesis 4 plus 2 right parenthesis space equals space minus 6 straight A subscript 22 space space equals space open vertical bar table row 2 cell space space space space space 1 end cell row 1 cell space space space 2 end cell end table close vertical bar space equals space 4 minus 1 space equals space 3 straight A subscript 23 space equals space minus open vertical bar table row 2 cell space space space space space space space 2 end cell row 1 cell space space minus 2 end cell end table close vertical bar space equals space minus left parenthesis negative 4 minus 2 right parenthesis space equals space 6 straight A subscript 31 space equals space open vertical bar table row 2 cell space space 1 end cell row 1 cell space space 2 end cell end table close vertical bar space equals space 4 minus 1 space equals space 3

straight A subscript 32 space equals space minus open vertical bar table row 2 cell space space space 1 end cell row cell negative 2 end cell cell space space space 2 end cell end table close vertical bar space equals space minus left parenthesis 4 plus 2 right parenthesis space equals space minus 6 straight A subscript 33 space equals space open vertical bar table row 2 cell space space space 2 end cell row cell negative 2 end cell cell space space space 1 end cell end table close vertical bar space equals space 2 plus 4 space space equals 6 therefore space space space adj. space straight A space equals space open square brackets table row 6 cell space space 6 end cell cell space space 3 end cell row cell negative 6 end cell cell space space 3 end cell cell space space 6 end cell row 3 cell space minus 6 end cell cell space space 6 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 6 cell space space minus 6 end cell cell space space 3 end cell row 6 cell space space space space 3 end cell cell negative 6 end cell row 3 cell space space space 6 end cell cell space space 6 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 space 1 over 27 open square brackets table row 6 cell space space minus 6 end cell cell space space 3 end cell row 6 cell space space space space space 3 end cell cell negative 6 end cell row 3 cell space space space space space 6 end cell cell space space 6 end cell end table close square brackets space space space space open vertical bar straight B close vertical bar space equals space open vertical bar table row 1 cell space space space space 3 end cell cell space space space space space space 2 end cell row 1 cell space space space space 1 end cell cell space space minus 1 end cell row 2 cell negative 3 end cell cell space space minus 1 end cell end table close vertical bar space equals space 1 open vertical bar table row 1 cell space space space 1 end cell row cell negative 3 end cell cell space minus 1 end cell end table close vertical bar minus 3 open vertical bar table row 1 cell space space space space space 1 end cell row 2 cell space minus 1 end cell end table close vertical bar plus 2 space open vertical bar table row 1 cell space space space space 1 end cell row 2 cell space space minus 3 end cell end table close vertical bar space space space space space space space space space space space equals space 1 left parenthesis negative 1 plus 3 right parenthesis space minus space 3 left parenthesis negative 1 minus 2 right parenthesis space plus space 2 left parenthesis negative 3 minus 2 right parenthesis space equals space 2 plus 9 minus 10 space equals space 1 not equal to 0

straight B subscript 11 space equals space open vertical bar table row 1 cell space space 1 end cell row cell negative 3 end cell cell space minus 1 end cell end table close vertical bar space equals space minus 1 plus 3 space equals space 2 straight B subscript 12 space equals space minus open vertical bar table row 1 cell space space space space space space space space 1 end cell row 2 cell space space space space minus 1 end cell end table close vertical bar space equals space minus left parenthesis negative 1 minus 2 right parenthesis space equals space 3 straight B subscript 13 space equals space open vertical bar table row 1 cell space space space space space space 1 end cell row 2 cell space space minus 3 end cell end table close vertical bar space equals space minus 3 minus 2 space equals space minus 5 straight B subscript 21 space equals space minus open vertical bar table row cell space space space 3 end cell cell space space space 2 end cell row cell negative 3 end cell cell space minus 1 end cell end table close vertical bar space equals space minus left parenthesis negative 3 plus 6 right parenthesis space equals space 3 straight B subscript 22 space equals space open vertical bar table row 1 cell space space space space space space 2 end cell row 2 cell space space minus 1 end cell end table close vertical bar space equals space minus 1 minus 4 space equals space minus 5 straight B subscript 23 space equals negative open vertical bar table row 1 cell space space space space space space 3 end cell row 2 cell space minus 3 end cell end table close vertical bar space space equals negative left parenthesis negative 3 minus 6 right parenthesis space equals space 9 straight B subscript 31 space equals space open vertical bar table row 3 cell space space 2 end cell row 1 cell space 1 end cell end table close vertical bar space equals space 3 minus 2 space equals space 1 straight B subscript 32 space equals space open vertical bar table row 1 cell space space space 2 end cell row 1 cell space space 1 end cell end table close vertical bar space space equals negative left parenthesis 1 minus 2 right parenthesis space equals space 1 straight B subscript 33 space equals space open vertical bar table row 1 cell space space space 3 end cell row 1 cell space space 1 end cell end table close vertical bar space equals space 1 minus 3 space equals space minus 2

therefore space space space space adj. space straight B space equals space open square brackets table row 2 cell space space space space space 3 end cell cell space space minus 5 end cell row cell negative 3 end cell cell space minus 5 end cell cell space space space space space 9 end cell row 1 cell space space space 1 end cell cell 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 2 cell space space space minus 3 end cell cell space space space space space space 1 end cell row 3 cell space space minus 5 end cell cell space space space space space space 1 end cell row cell negative 5 end cell cell space space space space 9 end cell cell space space minus 2 end cell end table close square brackets 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 open square brackets table row cell space space space space 2 end cell cell space space minus 3 end cell cell space space space 1 end cell row cell space space space space 3 end cell cell space space minus 5 end cell cell space space space 1 end cell row cell negative 5 end cell cell space space space space space space 9 end cell cell space space minus 2 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 1 over 27 open square brackets table row cell space space 2 end cell cell space space minus 3 end cell cell space space space space space space 1 end cell row cell space space space 3 end cell cell space minus 5 end cell cell space space space space space space 1 end cell row cell negative 5 end cell cell space space 9 end cell cell space space minus 2 end cell end table close square brackets space open square brackets table row 6 cell space space minus 6 end cell cell space space space space space 3 end cell row 6 cell space space space space space space 3 end cell cell space minus 6 end cell row 3 cell space space space space space 6 end cell cell space space space 6 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 1 over 27 open square brackets table row cell 12 minus 18 plus 3 end cell cell space space space space space minus 12 minus 9 plus 6 end cell cell space space space space space 6 plus 18 plus 6 end cell row cell 18 minus 30 plus 3 end cell cell space space space space space minus 18 minus 15 plus 6 end cell cell space space space space space 9 plus 30 plus 6 end cell row cell negative 30 plus 54 minus 6 end cell cell space space space space space 30 plus 27 minus 12 end cell cell space space space minus 15 minus 54 minus 12 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 1 over 27 open square brackets table row cell negative 3 end cell cell space space space space space minus 15 end cell cell space space space space 30 end cell row cell negative 9 end cell cell space space space space minus 27 end cell cell space space space 45 end cell row 18 cell space space space space 45 end cell cell space space minus 81 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... left parenthesis 1 right parenthesis Also comma space space space space AB space equals space open square brackets table row 2 cell space space 2 end cell cell space space space 1 end cell row cell negative 2 end cell cell space space 1 end cell cell space space space 2 end cell row 1 cell space space minus 2 end cell cell space space space 2 end cell end table close square brackets space open square brackets table row 1 cell space space space 3 end cell cell space space space space 2 end cell row 1 cell space space space space 1 end cell cell space space space space 1 end cell row 2 cell space space minus 3 end cell cell space space minus 1 end cell end table close square brackets space space space space space space space space

equals space open square brackets table row cell space 2 plus 2 plus 2 end cell cell space space space space space space 6 plus 2 minus 3 end cell cell space space space space space space space 4 plus 2 minus 1 end cell row cell negative 2 plus 1 plus 4 end cell cell space space minus 6 plus 1 minus 6 end cell cell space space space minus 4 plus 1 minus 2 end cell row cell space space 1 minus 2 plus 4 end cell cell space space space space 3 minus 2 minus 6 end cell cell space space space space space 2 minus 2 minus 2 end cell end table close square brackets space equals space open square brackets table row 6 cell space space space space 5 end cell cell space space space 5 end cell row 3 cell space minus 11 end cell cell space minus 5 end cell row 3 cell negative 5 end cell cell space minus 2 end cell end table close square brackets open vertical bar AB close vertical bar space equals space open square brackets table row 6 cell space space space 5 end cell cell space space space space space 5 end cell row 3 cell space minus 11 end cell cell space space minus 5 end cell row 3 cell space minus 5 end cell cell space space minus 2 end cell end table close square brackets space equals space 6 open vertical bar table row cell negative 11 end cell cell space space minus 5 end cell row cell negative 5 end cell cell space space minus 2 end cell end table close vertical bar space minus space 5 open vertical bar table row 3 cell space space space minus 5 end cell row 3 cell space space minus 2 end cell end table close vertical bar space plus space 5 space open vertical bar table row 3 cell space space minus 11 end cell row 3 cell space space minus 5 end cell end table close vertical bar space space space space space space space space space space space equals 6 left parenthesis 22 minus 25 right parenthesis minus 5 left parenthesis negative 6 plus 15 right parenthesis plus 5 left parenthesis negative 15 plus 33 right parenthesis space equals space minus 18 minus 45 plus 90 space equals space 27

Co-factors of the elements of first row of | AB | and –3, –9, 18 respectively. Co-factor of the elements of second row of | AB | and –15, –27, 45 respectively. Co-factors of the elements of third row of | AB | and 30, 45, –81, respectively.

therefore space space space adj. space left parenthesis AB right parenthesis space equals space open square brackets table row cell negative 3 end cell cell space space minus 9 end cell cell space space 18 end cell row cell negative 15 end cell cell space space minus 27 end cell cell space 45 end cell row 30 cell space space 45 end cell cell negative 81 end cell end table close square brackets space equals space open square brackets table row cell negative 3 end cell cell space space minus 15 end cell cell space space 30 end cell row cell negative 9 end cell cell space minus 27 end cell 45 row 18 45 cell negative 81 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 1 over 27 open square brackets table row cell negative 3 end cell cell space space minus 15 end cell cell space space 30 end cell row cell negative 9 end cell cell space minus 27 end cell cell space space 45 end cell row 18 cell space 45 end cell cell negative 81 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... left parenthesis 2 right parenthesis

From (1) and (2), we get,

(AB)^–1 = B^–1 A^–1.

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