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practice_midterm

# practice_midterm - A 3(15 points Suppose S is a 2 n × 2 n...

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Practice Midterm Exam (Math 2J: 44580) Instructor: Peng Song 1. Giving the following system: 2 x 1 + 3 x 2 + x 3 = 1 x 1 + x 2 + x 3 = 3 3 x 1 + 4 x 2 + 2 x 3 = 4 (1) (5 points). What are the coeFcient matrix and the augmented matrix of this system? (2) (10 points). Use Gaussian elimination to determine the solutions (if any) to the system. 2. Given a 3 × 3 matrix A : A = ( a ij ) 3 × 3 = 3 2 4 1 - 2 3 2 3 2 (1) (6 points). What are the minors of a 11 , a 12 , a 13 ? (2) (9 points). ±ind the values of the cofactors A 11 , A 12 , A 13 . (3) (10 points). ±ind the determinate of A : det( A ), and the adjoint of A : adj A . (4) (10 points). What is the inverse of

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Unformatted text preview: A ? 3 (15 points). Suppose S is a 2 n × 2 n matrix: S = b I n A O I n B where I n is the n × n identity matrix and A is an n × n matrix. Determine the block form of S-1 . 4 (10 points). If A is an n × n nonsingular matrix satisfying A 2 = A , show that A = I n . 1 5. Given the matrix A : A = b 0 1 1 0 B (1) (15 points). Find the eigenvalues and all corresponding eigenspaces of the matrix A . (2) (10 points). factor the matrix A into a product XDX-1 , where D is a diagonal matrix. 2...
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practice_midterm - A 3(15 points Suppose S is a 2 n × 2 n...

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