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20f-mt2-soln

# 20f-mt2-soln - MATH 20F MIDTERM 2 You have 50 minutes for...

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MATH. 20F MIDTERM 2 You have 50 minutes for this exam. Please write legibly and show all working. No calculators are allowed. Write your name and ID number. Name: ID Number: (1i) (6 points) Suppose that B is a matrix to which the following sequence of ERO’s is performed: (a) Exchange rows 1 and 3; (b) Add 3 times of Row 2 to Row 3; (c) Multiply row 1 by 2, yielding the matrix C = 1 0 0 2 1 0 9 3 1 . Find det B . ERO (a) multiplies det by 1; ERO (b) does not change det, and ERO (c) multiplies det by 2. Hence, det C ) = 2 · det B . On the other hand, det B = 1, so det C = 1 / 2. (ii) (4 points) If an n × n matrix A satisfies A T · A = I n , what can you say about det A ? We have: 1 = det I n = det( A T A ) = det( A T ) · det A = (det A ) 2 . So det A = ± 1 . 1

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2 MATH. 20F MIDTERM 2 (2) Let V = M 3 × 3 be the vector space of 3 × 3 matrices. Let W 1 = { A V : A is invertible } and W 2 = { A V : A T = A } (i) (10 points) Decide if W 1 and W 2 are subspaces of V . Justify your answer.
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