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# exam2 - MATH 20F SAMPLE MIDTERM 2 You have 50 minutes for...

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MATH. 20F SAMPLE 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) (5 points) Find the determinant of A = 1 0 0 1 2 0 1 0 11 3 7 5 0 0 - 1 0 . Calculate the determinant by using cofactor expansion along the 2nd col- umn and then the 3rd column: det A = - 3 · det 1 0 1 2 1 0 0 - 1 0 = - 3 · 1 · det p 2 1 0 - 1 P = ( - 3) · 1 · ( - 2) = 6 (ii) (5 points) If A is an n × n matrix, how are det(3 A ) and det( A ) related? The matrix 3 A is obtained from A by multiplying each row by 3. Each such ERO multiplies the determinant by 3. Hence det(3 A ) = 3 n · det( A ) . 1

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2 MATH. 20F SAMPLE MIDTERM 2 (2i) (10 points) Let P 2 be the vector space of polynomials of degree 2, and let W = { p P 2 : p (0) + p (1) = 0 } . Show that W is a subspace of P 2 and Fnd a basis for W . It is clear that the zero polynomial is in
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exam2 - MATH 20F SAMPLE MIDTERM 2 You have 50 minutes for...

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