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...

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 1 1 1 3 4 3 1 A , ? ? ? ? ? ? ? ? ? ? ? ? 5 0 3 1 2 5 B , and ? ? ? ? ? ? ? ? ? ? 0 6 7 1 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 ?C D . Demonstrate this property for the following four matrices: ? ? ? ? ? ? ? ? ? ? ? 3 5 4 2 7 1 A , ? ? ? ? ? ? ? ? ? ? ? 0 0 1 0 1 1 1 1 2 B , ? ? ? ? ? ? ? ? ? ? ? 4 5 6 2 3 0 1 0 0 C , and ? ? ? ? ? ? ? ? ? ? ? ? ? 11 2 5 0 3 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) B D ? ? (viii) C D ? ? (ix) B D ?C D ? ? (x) Your answers in (vi) and (ix) should agree. Math 1057—Fall 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 4 2 7 1 A , ? ? ? ? ? ? ? ? ? ? ? 0 0 1 0 1 1 1 1 2 B , ? ? ? ? ? ? ? ? ? ? ? 4 5 6 2 3 0 1 0 0 C , and ? ? ? ? ? ? ? ? ? ? ? ? ? 11 2 5 0 3 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 correct it. Provide a computer printout of both the commands and the output for each check. Scalar Multiplication Property 5 Any scalar can always be factored out of matrix multiplication. In other words, for any two matrices A and B of appropriate size and for any scalar (number) c , the following is true: c?AB? ? ?cA?B ? A?cB?. Demonstrate this fact for the following matrices: ? ? ? ? ? ? ? ? ? ? ? 3 5 4 2 7 1 A and ? ? ? ? ? ? ? ? ? ? ? ? ? 11 2 5 0 3 4 B (i) AB ? ? (ii) ? 3?AB? ? ? (iii) ?? 3A?B ? ? (iv) ? A?? 3B? ? ? (v) Your answers in (ii), (iii) and (iv) should agree.