Find the minors and co-factors of the elements a 11 , a 21 in the determinant. - Wired Faculty

Question

Find the minors and co-factors of the elements a₁₁, a₂₁ in the determinant.

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Solution

increment space equals space open vertical bar table row cell straight a subscript 11 end cell cell space straight a subscript 12 end cell cell space straight a subscript 13 end cell row cell straight a subscript 21 end cell cell space straight a subscript 22 end cell cell straight a subscript 23 end cell row cell straight a subscript 31 end cell cell space straight a subscript 32 end cell cell straight a subscript 33 end cell end table close vertical bar

increment space equals space open vertical bar table row cell straight a subscript 11 end cell cell space straight a subscript 12 end cell cell space straight a subscript 13 end cell row cell straight a subscript 21 end cell cell space straight a subscript 22 end cell cell space straight a subscript 23 end cell row cell straight a subscript 31 end cell cell space straight a subscript 32 end cell cell space straight a subscript 33 end cell end table close vertical bar

Minor of

straight a subscript 11 space equals space straight M subscript 11 space equals space open vertical bar table row cell straight a subscript 22 end cell cell space space space space straight a subscript 23 end cell row cell straight a subscript 32 end cell cell space space space straight a subscript 33 end cell end table close vertical bar space equals space straight a subscript 22 straight a subscript 33 minus straight a subscript 23 straight a subscript 32

Co-factor of

straight a subscript 11 space equals space straight A subscript 11 space equals space left parenthesis negative 1 right parenthesis to the power of 1 plus 1 end exponent straight M subscript 11 space equals space straight M subscript 11 space equals space straight a subscript 22 straight a subscript 33 minus straight a subscript 23 straight a subscript 32

Minor of

straight a subscript 21

equals straight M subscript 21 space equals space open vertical bar table row cell straight a subscript 12 end cell cell space space straight a subscript 13 end cell row cell straight a subscript 13 end cell cell space space straight a subscript 33 end cell end table close vertical bar space equals space straight a subscript 12 straight a subscript 33 space minus space straight a subscript 13 straight a subscript 32

Co-factor of

straight a subscript 21 space equals space straight A subscript 21 space equals space left parenthesis negative 1 right parenthesis to the power of 2 plus 1 end exponent space space straight M subscript 21 space equals space minus straight M subscript 21 space equals space minus straight a subscript 12 straight a subscript 33 plus straight a subscript 13 straight a subscript 32

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