VAM_32-43 - Tutorial V AM V ector Algebra and an...

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Tutorial VAM: Vector Algebra and an Introduction to Matrices M (12) = ¯ ¯ ¯ ¯ ¡ 1 1 2 2 ¯ ¯ ¯ ¯ = ( ¡ 1) £ 2 ¡ 1 £ 2 = ¡ 4 ; M (13) = ¯ ¯ ¯ ¯ ¡ 1 2 2 4 ¯ ¯ ¯ ¯ = ( ¡ 1) £ 4 ¡ 2 £ 2 = ¡ 8 = ) det M = 1 £ 0+ ( ¡ 1) £ 3 £ ( ¡ 4) +2 £ ( ¡ 8) = ¡ 4 : Exercise 33. The even permutations of f 123 g are: f 123 g ; f 231 g ; f 312 g : Note that, for the special case of 3 integers, these are obtained by starting with anyinteger in the original sequence and then reading left to right, cycling back to the beginning when the end of the sequence is reached. If these are numerals on a three-digit clock face, the even permutations are obtained by circling through them clockwise. In general, any even permutation can be obtained from any other even permutation by an even number of interchanges. The odd permutationsof f 123 g are: f 132 g ; f 213 g ; f 321 g : These are obtained by starting with any integerin the original sequence f 123 g and reading right to left, cycling back to the end when the beginning of the sequence isreached. They also correspond tocounterclockwise readingofnumbers on a three-digit clock face. Any odd permutation can be obtained from any even permutation by an odd numberofinterchanges, or from any other odd permutation by an even number of interchanges. The 12 even permutations of f 1234 g are: f 1234 g ; f 1342 g ; f 1423 g ; f 2143 g ; f 2314 g ; f 2431 g ; f 3124 g ; f 3241 g ; f 3412 g ; f 4132 g ; f 4213 g ; f 4321 g : The 12 odd permutations of f 1234 g are: f 1243 g ; f 1324 g ; f 1432 g ; f 2134 g ; f 2341 g ; f 2413 g ; f 3142 g ; f 3214 g ; f 3421 g ; f 4123 g ; f 4231 g ; f 4312 g : For n > 3 there is no simple way of relating the even- or oddness or a permutation to a simple cycle. One must count interchanges. Exercise 34. Consider, for example, expansion of a rank 3 determinant in minors about the first row. Then, from Eqn. 59, det M (3) = (+1) M 11 ( M 22 M 33 ¡ M 23 M 32 ) + ( ¡ 1) M 12 ( M 21 M 33 ¡ M 23 M 31 ) +(+1) M 13 ( M 21 M 32 ¡ M 22 M 31 ) = 3 X f klm g =1 P f 123 g M 1 k M 2 l M 3 m ; as can be seen by inspection. Now, assume that Eqn. 60 is true for a rank-
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