X Course Hero Differential Equations and Linear X G Determine all values of the const x M (no subject) - sylviaabdelmasih@ x ndl.ethernet.edu.et/bitstream/123456789/31552/1/Differential%20Equations%20and%20Linear%20Algebra.pdf k Hel... Dashboard MyMav - The Unive... Paraphrasing Tool |... Course Hero YouTube G Google Course Home Inbox (44) - sylvia.a... and Linear Algebra, 4th Edition 186 / 871 80% (c) A linear system Ax = b with an n x n invertible co- efficient matrix A has a unique solution. 9. A = AN - d) If A is a matrix such that there exists a matrix B with AB - In, then A is invertible. (e) If A and B are invertible n x / matrices, then so is 10. A 121 A+ B. 6 7 (f) If A and B are invertible n x n matrices, then so is O 10 AB. 1. A= 0 0 1 g) If A is an invertible matrix such that A- = A, then A is the identity matrix. 13 12. A - WNA NEN (h) If A is an n Xn invertible matrix and B and Care n x n matrices such that AB - AC, then B = C. (i) If A is a 5 x 5 matrix of rank 4, then A is not invertible. 6 AN (j) If A is a 6 x 6 matrix of rank 6, then A is invertible. Problems 14. A UNN For Problems 1-4, verify by direct multiplication that the N+ given matrices are inverses of one another. 15. A - 16. A NON! WVWN ANW W - N O - W N - W - N INON OHOH WIT ON - J M O pu provided ad - be # o . 17. A = AN OW - For Problems 5-18, determine A if possible, using the 18. A = JUO Gauss-Jordan method. If A- exists, check your answer by o verifying that AA = In oo 5. A= 1 3] 19. Let A 1 - w Find the third column vector of A without determining the other columns of the inverse matrix. 7. 4-[ -14 2]. 20. Let A = 5 i Find the second column 8. A =[: : ]. vector of A -I without determining the other columns of the inverse matrix. ASUS VivoBook F2 F3 F4 -O- F5 Fo F7 F8 F9 F11 F12 /A Prt Sc Insert 3 5 O E R U O P D F G H Kel... Dashboard MyMav - The Unive. Paraphrasing Tool |... * Course Hero YouTube Google P Course Home Inbox (44) - sylvia.a... Linear Algebra, 4th Edition 187 / 871 - 80% + U.A-1-1 1 of the inverse matrix. 7. A =[-141 72]. 20. Let A Find the second column Lool vector of A " without determining the other columns of the inverse matrix. 178 CHAPTER 2 Matrices and Systems of Linear Equations For Problems 21-26, use A to find the solution to the given 36. Let A be an n x / matrix with A" - 0. Prove that system. I, - A is invertible with 6x1 + 20x2- -8, 21. 2xi + 7x2- ( I, - A) - InTATATA 22. 1 + 3x2- 1, 37. Suppose the inverse of the matrix A is B3. What is the inverse of A ? Prove your answer. 2x1 + 5x2- 3. X1 + X2 2x3 - -2, 38. Suppose the inverse of the matrix A is B . What is 23. - 3. the inverse of A ? Prove your answer. 2 x1 + 4x2 - 305- 39. Prove that if A, B, C are n x n matrices satisfying 24. X1 -21x2- BA - In and AC - In, then B = C. (2 1)x1 + 4ix2 - 10. If A, B. C are n x n matrices satisfying BA = I, and 3x1 + 4x2 + 5.13 - 1, CA - In, does it follow that B - C? Justify your 25. 2x1 + 10x2 + 13 = 1, answer. 4x1 + 12 + 8 x3 = 1 . 41. Consider the general 2 x 2 matrix x1 + 12 + 2x3 - 12, 26. X1 + 2x2- X3 -24, A = 2x1 - 12 + X3= - 36. and let A - alla22 -a12a21 with all # 0. Show that An n x n matrix A is called orthogonal if A if A # 0, For Problems 27-30, show that the given matrices are orthogonal. 27. A = ] The quantity A defined above is referred to as the de terminant of A. We will investigate determinants in 28. A = 1/2 3/2 more detail in the next chapter. 29. A _ cosa sin a 12. Let A be an n x n matrix, and suppose that we have - sina cosa to solve the p linear systems - 2x Ax/ - b/, 30. A - 1+ 2x 2 - 2x i = 1. 2. ... . P where the b, are given. Devise an efficient method for 31. Complete the proof of Theorem 2.6.10 by verifying solving these systems. ASUS VivoBook F2 F3 FA F5 F7 F8 F9 F10 F11 F12 Prt Sc Insert -O & 3 5 8 O E R T U O P D F G H K C V B N M