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Question
Show with the usual notation that for any matrix A = `["a"_"ij"]_(3xx3) "is" "a"_11"A"_21 + "a"_12"A"_22 + "a"_13"A"_23 = 0`
Solution
A = `["a"_"ij"]_(3xx3) = [("a"_11,"a"_12,"a"_13),("a"_21,"a"_22,"a"_23),("a"_31,"a"_32,"a"_33)]`
`"A"_21 = (-1)^(2+1)"M"_21 = - |("a"_12,"a"_13),("a"_32,"a"_33)|`
`= - ("a"_12"a"_33 - "a"_13"a"_32)`
`= - "a"_12"a"_33 + "a"_13"a"_32`
`"A"_22 = (-1)^(2+2)"M"_22 = - |("a"_11,"a"_13),("a"_31,"a"_33)|`
= a11a33 - a13a31
`"A"_23 = (-1)^(2+3)"M"_23 = - |("a"_11,"a"_12),("a"_31,"a"_32)|`
= - (a11a32 - a12a31)
= - a11a32 + a12a31
∴ `"a"_11"A"_21 + "a"_12"A"_22 + "a"_13"A"_23`
`= "a"_11 (- "a"_12"a"_33 + "a"_13"a"_32) + "a"_12("a"_11"a"_33 - "a"_13"a"_31) + "a"_13(- "a"_11"a"_32 + "a"_12"a"_31)`
`= - "a"_11"a"_12"a"_33 + "a"_11"a"_13"a"_32 + "a"_11"a"_12"a"_33 - "a"_12"a"_13"a"_31 - "a"_11"a"_13"a"_32 + "a"_12"a"_13"a"_31`
= 0
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