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UCLA Linear Algebra Final

# UCLA Linear Algebra Final - MATH 33A Practice Final Exam...

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MATH 33A: Practice Final Exam Winter 2009 – Dr. Frederick Park Note: this exam is only slightly longer than the actual Final. 1. Find all 3 × 3 orthogonal matrices of the form A = a b 0 c d 1 e f 0 2. Find an orthogonal transformation T : R 3 R 3 such that T 2 / 3 2 / 3 1 / 3 = 0 0 1 3. For what values k is the following matrix invertible A = 1 k 1 1 k + 1 k + 2 1 k + 2 2 k + 4 4. Calculate the determinant of A = 1 1 0 1 1 0 2 0 7 0 2 3 4 5 0 0 0 0 3 0 3 4 5 2 6 1

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5. Use Gaussian Elimination to calculate the determinant of A = 0 0 0 0 2 1 0 0 0 3 0 1 0 0 4 0 0 1 0 5 0 0 0 1 6 6. Is the function T a b c d = ad + bc linear in the rows and columns of the matrix? 7. Use Cramer’s rule to solve the system 2 x + 3 y = 8 4 y + 5 z = 3 6 x + 7 z = - 1 8. Find and invertible matrix S and a diagonal matrix D such that S - 1 AS = D where A = 3 - 4 0 2 - 3 0 0 0 1 9. For which value of a is the matrix A = 0 0 a 1 0 0 0 1 0 2
diagonalizable?

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UCLA Linear Algebra Final - MATH 33A Practice Final Exam...

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