Suppose that A is a 3 × 3 matrix, A T + 5 I 3 is...

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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 7524 ItemID 3004 DownloaderID 7524 ItemID 3004 With dedication from a sharer @
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. 1 2 1 2 1 2 2 1 3 5 5 7 3 6 4 9 10 11 1 2 4 3 6 9 R 2 = R 2 2 R 1 -→ 1 2 1 2 1 2 0 - 3 1 1 3 3 3 6 4 9 10 11 1 2 4 3 6 9 R 3 = R 3 3 R 1 -→ 1 2 1 2 1 2 0 - 3 1 1 3 3 0 0 1 3 7 5 1 2 4 3 6 9 R 4 = R 4 R 1 -→ 1 2 1 2 1 2 0 - 3 1 1 3 3 0 0 1 3 7 5 0 0 3 1 5 7 R 4 = R 4 3 R 3 -→ 1 2 1 2 1 2 0 - 3 1 1 3 3 0 0 1 3 7 5 0 0 0 - 8 - 16 - 8 R 4 = 1 / 8 R 4 -→ 1 2 1 2 1 2 0 - 3 1 1 3 3 0 0 1 3 7 5 0 0 0 1 2 1 R 2 = 1 / 3 R 2 -→ 1 2 1 2 1 2 0 1 - 1 / 3 - 1 / 3 - 1 - 1 0 0 1 3 7 5 0 0 0 1 2 1 Since we have four leading ones and five variables, we will have one parameter. Let x 5 = t , where t is any real number. The fourth row implies that x 4 + 2 x 5 = 1 x 4 = 1 - 2 t. The third row implies that x 3 + 3 x 4 + 7 x 5 = 5 x 3 + 3(1 - 2 t ) + 7 t = x 3 + 3 + t = 5 x 3 = 2 - t. The second row implies that x 2 - 1 3 x 3 - 1 3 x 4 - x 5 = - 1 x 2 - 1 3 (2 - t ) - 1 3 (1 - 2 t ) - t = x 2 - 1 = - 1 x 2 = 0 . The first row implies that x 1 + 2 x 2 + x 3 + 2 x 4 + x 5 = 2 x 1 + (2 - t ) + 2(1 - 2 t ) + t = 2 x 1 = - 2 + 4 t. Hence, the general solution is x 1 = 4 t - 2 ,x 2 = 0 ,x 3 = 2 - t,x 4 = 1 - 2 t,x 5 = t, where t is any real number. 2 of 10 DownloaderID 7524 ItemID 3004 DownloaderID 7524 ItemID 3004 With dedication from a sharer @
EXTRA PAGE FOR QUESTION 1 - do not remove. 3 of 10 DownloaderID 7524 ItemID 3004 DownloaderID 7524 ItemID 3004 With dedication from a sharer @ Downloader ID: 7524 Item ID: 3004 Downloader ID: 7524 Downloader ID: 7524 Item ID: 3004 Downloader ID: 7524 Item ID: 3004 Item ID: 3004
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 . Item ID: 3004
4 of 10 ItemID 3004 With dedication from a sharer @ Item ID: 3004 Downloader ID: 7524 Downloader ID: 7524
EXTRA PAGE FOR QUESTION 2 - do not remove.

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