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assig9

# assig9 - counterexample 4 Consider the following matrix A =...

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Math 152, Spring 2010 Assignment #9 Notes: Each question is worth 5 marks. Due in class: Monday, March 22 for MWF sections; Tuesday, March 23 for TTh sections. Solutions will be posted Tuesday, March 23 in the afternoon. No late assignments will be accepted. 1. Find all the values of λ for which the matrix A = λ 0 2 - 1 λ - 1 0 - 1 λ - 2 is not invertible. 2. Consider the following linear system x + y = 1 3 x + 2 y + 1 z = - 1 x - 2 y + z = 5 (a) Write this system in the form A x = b for a matrix A and a vector b . (b) Compute det( A ). Is the matrix A invertible? Please explain. (c) In the case that A is invertible compute A - 1 . (d) Use the answer of part (c) to solve the original system. 3. Suppose that A and B are 3 × 3 matrices such that det ( A ) = 1 and det ( B ) = 2. Compute (a)-(d) below and consider case (e) for these matrices. (a) det(2 A ) (b) det( A T B )

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(c) det( A 100 B 3 ) (d) det( ABA - 1 ) (e) Is it necessarily the case that det( A + B ) = 3? If the answer is
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Unformatted text preview: counterexample. 4. Consider the following matrix: A = 1 2-1 4 2 0 1 2-2 0 0 1-1 0 0 3 4 5 1 0 1 2 3 (a) Compute the determinant of A . Hint: The computations become more manageable by expanding the determinant in an appropriate way. (b) Suppose you wanted to enter the matrix A in MATLAB. Consider starting with the zero matrix of the right size: A = zeros(5,5) Write the MATLAB commands that would enter the two non-zero rows in the third row of A . 5. Let T : R 2 → R 2 be the linear transformation that corresponds to the orthogonal projection to a unit vector a making an angle of θ = 60 degrees with respect to the x-axis. Find the determinant of the matrix representing T ....
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assig9 - counterexample 4 Consider the following matrix A =...

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