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Midtest

B w h e nca 5 detc c if d is obtained from a by

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( b ) W h e nC=A 5 , det(C) = . (c) If D is obtained from A by interchanging rows three and five, then det(D) = . (d) If E is obtained from A by replacing row four with negative two times row four, then det(E) = . (e) What can you say about the kernel of the linear transformation T: 6 6 defined by T( x )=A x ? Why??

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MidTest/MAS3105 Page 5 of 5 6. (10 pts.) Obtain an LU factorization for the matrix A below. 1- 2 1- 4 A= 3 - 600 -6 12 -3 12 7. (10 pts.) Suppose that the matrix A has the following LU factorization: 100  3 - 14- 8 - 710  0 - 721 5 32 5 1  001- 4 . (a) If b is an element of 3 , the problem of solving the equation A x = b may be handled by solving a pair of matrix equations in a particular order. Give the equations, complete with their coefficient matrices, in the correct order. Use all of the information at hand, but do not attempt to solve either equation. (b) Do the columns of A span 3 ? How do you know?? (c) Are the columns of A linearly independent? How do you know??
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b W h e nCA 5 detC c If D is obtained from A by...

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