Math 1057窶認all 2013 Assignment IV Due: Thurs, September 26 Page 1 of 2 Properties of Matrix Multiplication Associative Property 1 Matrix multiplication is associative. That is, for any three matrices A, B, and C of appropriate size, the following matrix multiplication equality is true: ?AB?C ? A?BC? . Demonstrate this property for the following three matrices: ????? ????? ? ? ? ? 2 3 11 1 34 3 1 A , ????? ????? ? ? 5 03 12 5 B , and ? ? ?? ? ? ?? ? ? 0 6 71 1 2 C . (i) AB ? ? (ii) ?AB?C ? ? (iii) BC ? ? (iv) A?BC? ? ? (v) Your answers in (ii) and (iv) should agree. Distributive Property 2 Matrix multiplication is distributive. That is, for any four matrices A, B, C and D of appropriate size, the following two matrix multiplication equalities are true: A?B ?C? ? AB ? AC and ?B ?C?D ? BD?CD. Demonstrate this property for the following four matrices: ? ? ?? ? ? ?? ? ? ? 3 5 42 7 1 A , ????? ????? ? 0 0 10 1 11 1 2 B , ????? ????? ? 4 5 62 3 01 0 0 C , and ????? ????? ? ? ? 11 25 03 4 D (o) ?B ?C? ? ? (i) A?B ?C? ? ? (ii) AB ? ? (iii) AC ? ? (iv) AB ? AC ? ? (v) Your answers in (i) and (iv) should agree. (vi) ?B ?C?D ? ? (vii) BD ? ? (viii) C D ? ? (ix) BD?C D ? ? (x) Your answers in (vi) and (ix) should agree.
Math 1057窶認all 2013 Assignment IV Due: Thurs, September 26 Page 2 of 2 Commutative Property 3 Matrix multiplication is not commutative, in general. That is, for any two matrices A and B, it does not necessarily follow that: B A ? AB . Demonstrate this fact for the following matrices: ? ? ?? ? ? ?? ? ? ? 3 5 42 7 1 A , ????? ????? ? 0 0 10 1 11 1 2 B , ????? ????? ? 4 5 62 3 01 0 0 C , and ????? ????? ? ? ? 11 25 03 4 D (i) AD ? ? (ii) D A ? ? (iii) AD D A ?? (iv) BC ? ? (v) C B ? ? (vi) BC C B ?? (vii) AB ? ? (viii) B A ? ? (ix) AB B A ?? 4 Double-check all your computations in #3 using Maple. Zero points if the Maple check does not agree with the hand-calculation. So, if your Maple check does not agree with your hand-computation, then find your error and…