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M310 Take Home Exam Due Date: Wed. Dec. 5 1. Let A be a 2 2 matrix dened by A = 3 0 0 2 ; and (x; y) satisfy an equation x2 + y2 = 1. If (x0; y0) is the image of (x; y) under the matrix A that is x0 y0 = Axy ; nd the equation of (x0; y0) and sketch its graph. 2. Find det(A ?? nIn) where A is an n n matrix whose entries are all 1 and In is the n n identity matrix. 3. Use the determinant properties to simplify the given matrix and show that detA = (x ?? y)(x ?? z)(x ?? w)(y ?? z)(y ?? w)(z ?? w) for A = 2664 1 x x2 x3 1 y y2 y3 1 z z2 z3 1 w w2 w3 3775 4.

M310 Take Home Exam Due Date: Wed. Dec. 5 1. Let A be a 2 2 matrix dened by A = 3 0 0 2 ; and (x; y) satisfy an equation x2 + y2 = 1. If (x0; y0) is the image of (x; y) under the matrix A that is x0 y0 = Axy ; nd the equation of (x0; y0) and sketch its graph. 2. Find det(A ?? nIn) where A is an n n matrix whose entries are all 1 and In is the n n identity matrix. 3. Use the determinant properties to simplify the given matrix and show that detA = (x ?? y)(x ?? z)(x ?? w)(y ?? z)(y ?? w)(z ?? w) for A = 2664 1 x x2 x3 1 y y2 y3 1 z z2 z3 1 w w2 w3 3775 4. Let P(x1; y1) and Q(x2; y2) be two points in the plane. Show that the equation of the line through P and Q is given by det(A) = 0 where A = 24 x y 1 x1 y1 1 x2 y2 1 35 5. Suppose that S = fv1; v2; v3g is a linearly independent set of vectors in a vector space V. Is T = fw1;w2;w3g where w1 = v1 + v2 w2 = v1 + v3 w3 = v2 + v3 linearly dependent or linearly independent? Justify your answer.

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