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Matrix Arithmetic

Part 1: Basic Properties

Enter the matrices A, B, and H defined in your worksheet.

  1. Compute the product AB, also compute BA, and compare your answers. What can you deduce from this comparison?

  2. Compute the products A(BH) and (AB)H. Compare your answers. What property of matrix multiplication does this illustrate?

Do you think the property you just described holds for all matrices that are of appropriate sizes? Before you answer, we'll try another multiplication that cannot be "rigged" in advance. We'll make the same computations with matrices that have random integer entries. (Do this several times to see what happens.)

  1. Using the random matrix generator, construct three matrices: Make S a 3 x 4 matrix, T a 4 x 2 matrix, and U a 2 x 5 matrix. Then compute S(TU) and (ST)U. Re-enter several times, starting from the definitions of S, T, and U. Compare the values of S(TU) and (ST)U each time, and then answer the question about how these products compare.

  2. Using the matrices A, B, and H created in step 1, compute A(B+H) and AB+AH. Compare the answers. What property of matrix arithmetic have you illustrated?

  3. Using the matrices A and H created in step 1, solve the following matrix equation for P: 5A+2P=3H.
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