Expansion by Alien Cofactors What happened if we multiply cofactors by the wrong (alien) row? If A= (aij)33then expanding along row 2 gives Replacing the elements from row 2 with another row, say row 1, is equivalent to having a matrix with two equal rows. 18 212122222323a Ca Ca CA111112133132312313aaaaaaaaa
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Expansion by Alien Cofactors We know: Theorem 16.5.1 Let A= (aij)nnwith cofactors CijThen: (i) (ii) and similarly for expanding along columns. 19 1121122213230a Ca Ca C1122iiiiinina Ca Ca CA1122if , 0ikikinknkia Ca Ca C
Deriving formula: Let A= (aij)nnwith cofactors CijEquivalently: 20 11()||adjAΑA1122, if 0, if ikikinknika Ca Ca Cik A211212222221211121211121000000nnnnnnnnnnnnCCaaaCCaaaaCCaaCCC AAA