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midterm1review

# midterm1review - 7 Consider the matrix A = 1 2 2 3 1-1...

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Practice Problems for Midterm 1 1. Consider the following linear system x - y + z = 1 4 x + y + z = 5 2 x +3 y - z = c . (a) Give the augmented matrix of the system and transform it to a reduced row echelon form. (b) Decide how many solutions the system has according to the value of a . (c) In the case that the system has inﬁnitely many solutions, ﬁnd a general solutions in a parameterized form. 2. Consider the matrix A = 1 2 0 3 2 0 1 0 3 2 0 - 1 2 . (a) Find A - 1 . (b) Find A 3 . 3. Consider two invertible matrices A = 0 0 1 2 1 0 0 0 1 3 0 , B = 1 3 - 1 2 0 1 5 2 0 . (a) Find A - 1 . (b) Suppose that there is another matrix C satisfying BC = A . Find C - 1 . 4. Let A and B be symmetric n × n matrices. Also let C = AB . Show that AB = BA if C is symmetric. 5. Consider n × n invertible matrix A . Prove the following two statements. (a) ( A T ) - 1 = ( A - 1 ) T . (b) A 2 = I if and only if A = A - 1 . 6. Facor the matrix A = 2 0 1 - 4 1 0 6 3 1 into a LU formation.

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Unformatted text preview: 7. Consider the matrix A = 1 2 2 3 1-1 1-2-2 1 3 1 1 2 1 . Evaluate det( A ), det( A 2 ) and det( A-1 A T ). Answers 1.(a) 1 0 2 5 0 1-3 5 0 0 ± ± ± ± ± ± ± 6 5 1 5 c-3 (b) If c = 3, there are inﬁnitely many solution. If c 6 = 3, the system is inconsistent and there is no solution. (c) x =-2 5 α + 6 5 y = 3 5 α + 1 5 z = α . 2. (a) the same as A . (b) the same as A . 3. (a) 0 1 0 0 0 3 2 0 0 . (b) 2 1 15 6 2 6-2 . 6. 1 0 0-2 1 0 3 3 1 2 0 1 0 1 2 0 0-8 7. det( A )=4, det( A 2 )=16 and det( A-1 A T )=1...
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midterm1review - 7 Consider the matrix A = 1 2 2 3 1-1...

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