Ii 1 0 0 2 0 1 0 0 0 0 1 1 iii 1 0 0 7 0 1 0 3 0 0 0

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II. 1 0 0 2 0 1 0 0 0 0 1 1 III. 1 0 0 7 0 1 0 3 0 0 0 5 (b) ( 5 points ) For the system that has infinitely many solutions, please write the general solution to the sys- tem. Solve for the pivot variable(s) in terms of the free variable(s). 5. ( 3 points each ) For each of the following systems, state the elementary row operation that should come next when performing Gauss-Jordan elimination. Express the row operation in terms of R 1 , R 2 , R 3 . Do not perform the operation. (a) 1 - 5 1 - 2 4 6 (b) 1 4 - 2 5 0 - 3 6 9 0 4 3 1 6. ( 4 points each ) Let A = 1 - 1 3 2 and B = 4 0 - 2 1 . Perform the indicated operations. (a) A + B (b) 3 · A (c) A · B 1
7. ( 3 points each ) Let A = - 1 0 4 2 (a) AB (b) BC (c) CA 8. Find the inverse of each matrix. 1 3 , B = 6 0 0 1 5 7 , and C = 1 - 5 4 3 7 - 2 6 0 . For each of the following, is the product defined? If so, give the size of the product matrix ( do not multiply ). If not, state or show why it is not defined. ( 6 points ) ( 10 points ) (a) A = 1 - 3 2 - 4 (b) B = 1 0 - 4 1 1 3 - 2 0 0 1 9. (a) ( 8 points ) For the following system of linear equations, identify the matrices A, X, and B, and write the corresponding matrix equation . ( 2 x + 2 y = 4 1 2 x + y = 1 (b) ( 8 points ) Use the fact that 2 2 1 2 1 - 1 = 1 - 2 - 1 2 2 to solve the system. 2

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