Consider the system of linear equations x 1 +...

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University of Toronto Department of Mathematics MAT223H1F Linear Algebra I Midterm Examination October 14, 2010 H. Kim, F. Murnaghan, B. Rowe, S. Uppal Duration: 1 hour 20 minutes Last Name: Given Name: Student Number: Tutorial Code: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark 1 /10 2 /10 3 /10 4 /10 5 /5 6 /5 TOTAL /50 1 of 10 DownloaderID 21140 ItemID 3004
1. Consider the system of linear equations x 1 + 2 x 2 + x 3 + 2 x 4 + x 5 = 2 2 x 1 + x 2 + 3 x 3 + 5 x 4 + 5 x 5 = 7 3 x 1 + 6 x 2 + 4 x 3 + 9 x 4 + 10 x 5 = 11 x 1 + 2 x 2 + 4 x 3 + 3 x 4 + 6 x 5 = 9 . Find all solutions to the system and express your answer in parametric form.
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2. Suppose that A is a 3 × 3 matrix, A T + 5 I 3 is nonsingular (invertible) and ( A T + 5 I 3 ) 1 = 1 3 0 0 - 1 0 2 2 1 . Find A . First, find 1 3 0 0 - 1 0 2 2 1 1 using Gaussian elimination: 1 3 0 1 0 0 0 - 1 0 0 1 0 2 2 1 0 0 1 R 3 = R 3 2 R 1 -→ 1 3 0 1 0 0 0 - 1 0 0 1 0 0 - 4 1 - 2 0 1 R 2 = R 2 -→ 1 3 0 1 0 0 0 1 0 0 - 1 0 0 - 4 1 - 2 0 1 R 3 = R 3 +4 R 2 -→ 1 3 0 1 0 0 0 1 0 0 - 1 0 0 0 1 - 2 - 4 1 R 1 = R 1 3 R 2 -→ 1 0 0 1 3 0 0 1 0 0 - 1 0 0 0 1 - 2 - 4 1 Hence, we have A T + 5 1 0 0 0 1 0 0 0 1 = 1 3 0 0 - 1 0 - 2 - 4 1 A T = 1 3 0 0 - 1 0 - 2 - 4 1 - 5 0 0 0 5 0 0 0 5 = - 4 3 0 0 - 6 0 - 2 - 4 - 4 A = - 4 3 0 0 - 6 0 - 2 - 4 - 4 T = - 4 0 - 2 3 - 6 - 4 0 - 4 - 4 4 of 10 DownloaderID 21140 ItemID 3004 Downloader ID: 21140
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