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Course 111: Algebra, 14th Feb 2007 To be handed in at tutorials on Feb 19th and 20th. 1. Solve the following system of linear equations using Gauss-Jordan elimination. 2x1 + x2 − x3 = 2 x1 − x 2 + x 3 = 7 2x1 + 2x2 + x1 = 4 What is the rank of the coefficient matrix describing this system? Write the augmented matrix, Ag and perform row operations to transform it to reduced row-echelon form: 2 1 −1 2 Ag = 1 −1 1 7 2 2 1 4 e.r.o0 s → 1 0 0 3 0 1 0 −2 0 0 1 2 Where the row operations were 21 R1 , R2 − R1 , R3 − 2R1 , − 32 R2 , R3 − R2 , 13 R3 , R2 + R3 , R1 + 12 R3 , R1 − 12 R2 . Giving solution: x1 = 3, x2 = 2, x3 = 2. The rank of A is 3. 2. Compute the inverses of the following matrices −1 3 0 0 0 2 0 9 5 4 Using the row operations: 0 0 0 1 = 0 2 − 34 − 58 1 R , 1 R , R3 3 1 2 2 −1 1 3 1 −2 0 −2 3 1 0 1 0 (1) 1 4 − 9R1 , R3 − 5R2 , 41 R3 . 1 0 2 = 0 0 1 2 1 1 (2) Using the row operations: R2 + 2R1 , R2 ↔ R3 , R1 + 2R2 , R3 + R1 . −1 1 −1 3 1 2 2 −2 −2 1 − 31 2 = − 15 2 5 2 3 1 15 − 15 0 0 0) (3) Using the row operations: R2 −2R1 , R3 +2R1 , R3 −4R1 , 31 R2 , − 15 R3 , R1 + R2 , R1 − 53 R3 , R2 + 34 R3 . Given the matrix −2 0 −4 1 D = 1 0 −1 6 8 2 1 −1 Determine D T D. DT = −2 1 8 0 0 2 −4 −1 1 1 6 −1 (4) Therefore the product D T D is DT D = −2 1 8 0 0 2 −4 −1 1 1 6 −1 69 16 15 −4 −2 0 −4 1 16 4 2 −2 1 0 −1 6 = 15 2 18 −11 8 2 1 −1 −4 −2 −11 38 (5) Note that the product is symmetric even though D was not symmetric (or even square!).