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Sp06PracTest2

# Sp06PracTest2 - 1 2 x 1 4 x 2 3 x 3 3 x 4 = 3 x 1 x 2 x 3 x...

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Matrices and Matrix Calculations Spring 06, Unique Number: Practice TEST # 2 Problem 1. What is a basis for a subspace? (give a definition) (5 points) We consider the matrix B = 1 2 3 5 7 9 3 3 4 . Determine if the columns of B provide a basis for R 3 . (5 points) If they do give the coordinates of b = 2 1 1 in this basis. (5 points) Is the matrix B invertible and why? (5 points) Problem 2. We consider the matrix A = 2 4 0 6 1 2 0 3 3 1 5 - 1 2 - 2 6 - 6 - 1 3 - 5 7 . Is Col( A ) contained in R 4 or in R 5 ? (5 points) Give a basis A for Col( A ). (10 points) What is the rank of A ? (5 points) Give the coordinates of each column of A in the basis A (10 points) Problem 3. Give an LU factorisation of the matrix M = 1 2 3 1 1 3 3 2 2 4 3 3 1 1 1 1 . (10 points) What are the determinants of L and U ? Determine the value of det M without calculation . (5 points) Use the above factorization to solve the linear system: x 1 + 2 x 2 + 3 x 3 + x 4 = 2 x 1 + 3 x 2 + 3 x 3 + 2 x 4 = 1 2 x 1 + 4 x 2 + 3 x 3 + 3 x 4 = 3 x

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Unformatted text preview: 1 2 x 1 + 4 x 2 + 3 x 3 + 3 x 4 = 3 x 1 + x 2 + x 3 + x 4 =-1 (10 points) Problem 4. By using the Cramer’s rules solve the linear system: x 1 + x 2 + x 3 = 2 x 1 + x 2-x 3 = 3 x 1-x 2-x 3 = 4 No other method will be accepted (10points = 5 for Cramer’s formulas + 5 for calculations) 1 Problem 5. 1) Compute the determinant of the matrix: A = 3 2-1 1 6 3 2-4 (5 points). Is A invertible and why? (3 points) 2) Give the matrix of cofactors for A (5 points). What is the adjoint or adjugate matrix? (3 points) give the adjugate matrix of A (2 points). What is the inverse of A (formula: 5 points, calculations: 2 points). 3) Use the inverse of A to solve: ( 5 points.) 3 x 1 + 2 x 2-x 3 = 6 x 1 + 6 x 2 + 3 x 3 = 30 2 x 1-4 x 2 = 8 2...
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