B b 1 b 2 ab b write a 1 as a product of elementary

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B B 1 B 2 AB (b) Write A -1 as a product of elementary matrices. (Hint: You will need the inverses of the factors of A in the correct order.) A -1 =
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MidTest/MAS3105 Page 3 of 5 3. (18 pts.) (a) Write the general solution of the system of equations x 1 - 6x 2 + 8x 3 = 5 x 1 - 6x 2 - 8x 3 = -5 in parametric form. (b) If A = 1 -6 8 1 -6 -8 , Nul(A) is a subspace of k for which k ? k = (c) Using your results from part (a), give an explicit description of Nul(A). (d) Is the linear transformation T: 3 2 defined by T( x ) = A x , where A is as in part (b), onto? Explain. (e) When T is the linear transformation of part (d), what is the kernel, ker(T), of T ? Use all the information you have at hand. ker(T) = (f) What can you say about the column space of the matrix A of part (b)? Why? Explain as completely as possible. Col(A) =
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MidTest/MAS3105 Page 4 of 5 4. (10 pts.) (a) For which value(s) of s does the following equation, have a unique solution ? s 1 -2 x 1 1 0 s-2 1 x 2 = -2 0 0 3s x 3 3 (b) When the equation in (a) has a unique solution, obtain x 2 in terms of s when x is that unique solution. (Hint: Cramer’s rule might be simple enough here.) 5. (10 pts.) Suppose that A is a 6 x 6 matrix with det(A) = -10.
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