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Math 369 HW #3 Due at the beginning of class on Friday, September 19 Reading: Sections MISLE, LC, SS. Problems: 1. Let A= −1 2 −4 3 andB = 3/5 4/5 −2/5 −1/5 (a) Show that A and B are inverses of eachother. (b) Solve the following system of equations by using B appropriately (i.e., not by rowreducing): −x + 2y = 1 −4x + 3y = −2 * 2. Does the matrix have an inverse? If so, find it. If not, find some vector x which is not the * * zero vector so that A x = 0. (a) 1 2 −2 −1 5 −3 5 −4 0 (b) 1 0 2 0 3 0 3 0 8 a b ∈ Mat2,2 (R) and define the matrix 3. (a) Let A = c d d −b ∗ A = . −c a Show that AA∗ = A∗ A = (ad − bc)12 . (b) If ad − bc 6= 0, then the matrix 1 B= A∗ = ad − bc d ad−bc −c ad−bc makes sense. Show that B = A−1 . 4. Suppose B, C ∈ Matn,n (R) are both invertible. (a) Prove that (BC)−1 = C −1 B −1 . 1 −b ad−bc a ad−bc (b) Find specific 2 × 2 matrices B and C so that (BC)−1 6= B −1 C −1 . 5. Use Sage to help you solve the following system of equations: x1 −x1 x1 −x1 2x1 + − + + + 2x2 2x2 x2 5x2 2x2 + + + + + 3x3 3x3 x3 3x3 2x3 + − + − + 4x4 4x4 x4 2x4 3x4 + 5x5 + 5x5 + x5 + x5 + 3x5 + 6x6 − 6x6 + x6 − x6 + 3x6 = = = = = 7 7 2 1 4 (Hint: remember, you can create your own Sage worksheets on the Sage Math Cloud: http://sagemath.com. Also, clicking the little printer button on a Sage worksheet on the Sage Math Cloud will generate a PDF which you can then print or email.) 2