Linear Algebra with Applications (3rd Edition)

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Math 417 homework 4 solutions (Given only for problems that are not straightforward computation; contact the instructor if you still have questions about the others.) Section 2.3 problem 4,5,20 To invert a matrix A , bring h A . . . I n i to reduced row echelon form. If it is h I n . . . B i , then A is invertible and B = A - 1 . If you get h C . . . D i with C 6 = I n , then A is not invertible. 1 2 1 1 3 2 1 0 1 - 1 = 3 2 - 1 1 2 1 2 0 - 1 2 - 3 2 1 1 2 . The matrix of problem 5 is not invertible. 4 times its first column is the sum of second and third column (this is a nontrivial linear relation). The matrix of the transformation in problem 20 is 1 3 3 1 4 8 2 7 12 . Its inverse is - 8 - 15 12 4 6 - 5 - 1 - 1 1 . 1
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Section 2.4 problem 18 This is always true. ( A 2 ) · (( A - 1 ) 2 ) = AAA - 1 A - 1 = AIA - 1 = AA - 1 = I, so A 2 is invertible and ( A - 1 ) 2 is its inverse. It is customary to write this shorthand as A - 2 . Section 2.4 problem 20 This is sometimes but not always true.
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