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