Stitz-Zeager_College_Algebra_e-book

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Unformatted text preview: ed to meet the needs of a daily diet which requires 2500 calories, 1000 grams of protein, and 400 milligrams of Vitamin X. Now use Cramer’s Rule to ﬁnd the number of servings of Misty Mushrooms required. Does a solution to this diet problem exist? 10. Let R = −7 11 3 2 , S= 1 −5 6 9 T= 11 −7 2 3 , and U = −3 15 69 (a) Show that det(RS ) = det(R) det(S ) (b) Show that det(T ) = − det(R) (c) Show that det(U ) = −3 det(S ) 11. For M and N below, show that det(M ) = 0 and det(N ) = 0. 123 123 M = 1 2 3 , N = 4 5 6 456 000 12. Let A be an arbitrary invertible 3 × 3 matrix. (a) Show that det(I3 ) = 1.8 (b) Using the facts that AA−1 = I3 and det(AA−1 ) = det(A) det(A−1 ), show that det(A−1 ) = 8 1 det(A) If you think about it for just a moment, you’ll see that det(In ) = 1 for any natural number n. The formal proof of this fact requires the Principle of Mathematical Induction (Section 9.3) so we’ll stick with n = 3 for the time being. 8.5 Determinants and Cramer’s Rule 519 13. The purpose of this exercise is to introduce you to the eigenvalues and eigenvectors of a matrix.9 We begin with an example using a 2 × 2 ma...
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