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Determinants

Question
CBSEENMA12034519
Wired Faculty App

Find the minors and cofactors of the elements of the determinant:
open vertical bar table row 2 cell space space space minus 3 end cell cell space space space 5 end cell row 6 cell space space 0 end cell cell space space space 4 end cell row 1 cell space space 5 end cell cell negative 7 end cell end table close vertical bar
and verify that a11A31 + a13A33 = 0.

Solution
Let space increment space equals space open vertical bar table row 2 cell space space minus 3 end cell cell space space 5 end cell row 6 cell space space 0 end cell cell space space 4 end cell row 1 cell space space 5 end cell cell space space minus 7 end cell end table close vertical bar straight M subscript 11 space equals space open vertical bar table row 0 4 row 5 cell negative 7 end cell end table close vertical bar space equals space 0 minus 20 space equals space minus 20 comma space 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 minus 20 straight M subscript 12 space equals space open vertical bar table row 6 cell space space space space 4 end cell row 1 cell space space space minus 7 end cell end table close vertical bar space equals space minus 42 minus 4 space equals space minus 46 comma space space straight A subscript 12 space equals space left parenthesis negative 1 right parenthesis to the power of 1 plus 2 end exponent space straight M subscript 12 space equals space minus straight M subscript 12 space equals space 46 straight M subscript 13 space equals space open vertical bar table row cell 6 space end cell cell space space 0 end cell row 1 cell space space 5 end cell end table close vertical bar space equals space 30 minus 0 space equals space 30 comma space space straight A subscript 13 space equals space left parenthesis negative 1 right parenthesis to the power of 1 plus 3 end exponent space space space straight M subscript 13 space equals space straight M subscript 13 space equals space 30 straight M subscript 21 space equals space open vertical bar table row cell negative 3 end cell cell space space space space space 5 end cell row 5 cell space space minus 7 end cell end table close vertical bar space equals space 21 minus 25 space equals negative 4 comma space space straight A subscript 21 space equals space left parenthesis negative 1 right parenthesis to the power of 2 plus 1 end exponent straight M subscript 21 space equals space minus straight M subscript 21 space equals space 4 straight M subscript 22 space equals space open vertical bar table row 2 cell space space space space space space space space 5 end cell row 1 cell space space space space minus 7 end cell end table close vertical bar space equals space minus 14 minus 5 space equals space minus 19 comma space straight A subscript 22 space equals space left parenthesis negative 1 right parenthesis to the power of 2 plus 2 end exponent straight M subscript 22 space equals space straight M subscript 22 space equals space minus 19 straight M subscript 23 space equals space open vertical bar table row 2 cell space space space space minus 3 end cell row 1 cell space space space space space space space 5 end cell end table close vertical bar space equals space 10 plus 3 space equals space 13 comma space space space straight A subscript 23 space equals space left parenthesis negative 1 right parenthesis to the power of 2 plus 3 end exponent straight M subscript 23 space equals space minus straight M subscript 23 space equals space minus 13 straight M subscript 31 space equals space open vertical bar table row cell negative 3 end cell cell space space space 5 end cell row 0 cell space space space space space 4 space end cell end table close vertical bar space equals space minus 12 minus 0 space equals space minus 12 comma space space straight A subscript 31 space equals space left parenthesis negative 1 right parenthesis to the power of 3 plus 1 end exponent straight M subscript 31 space equals space straight M subscript 31 space equals space minus 12 straight M subscript 32 space equals space open vertical bar table row 2 cell space space space 5 end cell row 6 cell space space space space 4 end cell end table close vertical bar space equals 8 minus 30 space equals space minus 22 comma space space space straight A subscript 32 space equals space left parenthesis negative 1 right parenthesis to the power of 3 plus 2 end exponent straight M subscript 32 space equals space minus straight M subscript 33 space equals space 22 straight M subscript 33 space equals space open vertical bar table row 2 cell space space minus 3 end cell row 6 cell space space space space space space 0 end cell end table close vertical bar space equals space 0 plus 18 space equals space 18 comma space space space straight A subscript 33 space equals space left parenthesis negative 1 right parenthesis to the power of 3 plus 3 end exponent straight M subscript 33 space equals space straight M subscript 33 space equals space 18 Now space straight a subscript 11 straight A subscript 31 plus straight a subscript 12 straight A subscript 32 plus straight a subscript 13 straight A subscript 13 space equals space space space left parenthesis 2 right parenthesis thin space left parenthesis negative 12 right parenthesis plus left parenthesis negative 3 right parenthesis thin space left parenthesis 22 right parenthesis space plus space left parenthesis 5 right parenthesis thin space left parenthesis 18 right parenthesis 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 equals negative 24 minus 66 plus 90 space equals space 0.

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