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Unformatted text preview: 1.4 MATRIX ALGEBRA KIAM HEONG KWA pp. 44-45 (Theorem 1.4.1) For any scalars and and any matrices A , B , and C for which the indicated operations are defined, (1) A + B = B + A (2) ( A + B ) + C = A + ( B + C ) Notation: A + B + C = ( A + B ) + C (3) ( AB ) C = A ( BC ) Notation: ABC = ( AB ) C , A 2 = AA and A k = A k- 1 A for each integer k 2 (4) A ( B + C ) = AB + AC (5) ( A + B ) C = AC + BC (6) ( ) A = ( A ) Notation: A = ( ) A (7) ( AB ) = ( A ) B = A ( B ) Notation: AB = ( AB ) (8) ( + ) A = A + A (9) ( A + B ) = A + B Warning! Matrix multiplication is not commutative: in general, AB 6 = BA for most matrices A and B . Warning! Matrix multiplication does not obey the cancellation law: in general, AC = BC does not imply that A = B . Likewise, AB = AC does not imply that B = C generally. Example 1 (Exercise 1.4.4 in the text) . Let A ( , ) = 1- 1- 1 and let C = a b a b , where ,,,a,b are arbitrary scalars, such that A 6 = 0 and C 6 = 0 . Then A ( , ) C = C for any values of and . Letting A = A (0 , 1) = 1 1 and B = A (1 , 0) =...
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