1.8. MATRIX INVERSE 1
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1 .8. MTRIX INVERSE.8 Matrix Inverse MTH 9 SPRING 98 PRELIM # 9SP8PQ.tex.8. Let a) Find. Us any method you wish. check your result. b) Use your inverse above to nd all solutions to x = b; where b = MTH 9 SPRING 98 PRELIM # 9SP8PQ.tex.8. a) consider the x matrix B = k : : Find all values of k for which B has no inverse. b) In the vector space of polynomials of degree two or less are the vectors f x; + x ; x x g linearly independent? Give a reason. (You may assume that f; x; x g are linearly independent.) c) In the vector space S of solutions to the dierential equation y" y + y = re the vectors y = e t ; y = e t e t ; and y = e t linearly independent? Give a reason.
2 MTH 9 SPRING 98 FINL # 9SP8FQ.tex.8. a) Find a basis for the vector space of all x matrices. b) is the matrix given below, v is an eigenvector of. Find any eigenvalue of with v = [an eigenvector of ] = 6 c) Find one solution to each system of equations below, if possible. If not possible, explain why not. 6 x = b; b = 6 and b = 6 d) Read carefully. Solve for x in the equation b = x with: e) Find the inverse of the matrix and b = : MTH 9 FLL 98 FINL # 9F8FQ.tex.8. a) Find the eigenvalues and eigenvectors of the b) Find and T. c) Find the general solution of d dx y y y y MTH 9 SPRING 98 FINL # 9SP8FQ.tex.8. For the case ~ b = ~, the vector ~x = ~ a) Is always a solution. b) May or may not be a solution depending on. c) Is always the only solution. d) Is never a solution.
3 .8. MTRIX INVERSE MTH 9 FLL 98 FINL # 9F8FQ.tex.8.6 Find an elementary matrix E so that E is in reduced form, MTH 9 FLL 98 FINL # 9F8FQ.tex.8. Either compute the inverse of the matrix 6 ; or show that no inverse exists. MTH 9 SUMMER 989 PRELIM # 9SU89PQ.tex.8.8 Let be a x matrix with real entries. ssume that has an inverse Q and : Q = a) Find all vectors ~x in < such that ~x = b) Find the matrix. MTH 9 SPRING 99 PRELIM # 9SP9PQ.tex.8.9 Here a) Find. b) Find Xif X = c) Find ~vif ~v =.. MTH 9 SPRING 99 PRELIM # 9SP9PQ.tex.8. Given the 9 a) Find det. b) Find and check your answer. :
4 MTH 9 SPRING 99 FINL # 9SP9FQ.tex.8. a) Find the general solution, and write your answer as a particular solution plus the general solution of the associated homogeneous system. x y + z = x y + z = x + y + z = b) Check your answer for part a. c) Find the inverse of the d) Check your answer for part MTH 9 SPRING 99 PRELIM # 6 9SP9PQ6.tex.8. a) Find the inverse of the matrix b) Explain what you do to check your result in part (a), and then do it. c) Compute the determinant of the matrix () ; writing your result as a function of. d) Partially check your result by computing the determinant of (), and compare this value with the value of the function you found in (c) when =. MTH 9 FLL 99 PRELIM # 9F9PQ.tex.8. a) Find b) Use the result of part (a) to solve ~x = ~ b where ~ b c) Suppose B and C are N x N invertible matrices and you know B and C. What is (BC) equal to?
5 .8. MTRIX INVERSE MTH 9 FLL 99 FINL # 6 9F9FQ6.tex.8. The inverse of the matrix is a) b) c) d) e) MTH 9 SPRING 99 FINL # 9SP9FQ.tex.8. Give a formula for a b c d when it exists, and prove that your answer is correct. MTH 9 SPRING 99 PRELIM # 9SP9PQ.tex.8.6 a) Find the inverse of the matrix : b) Check your result by computing. c) Use the result to nd the solution of x y + z = x + y + z = x + y =
6 6 MTH 9 FLL 99 PRELIM # 9F9PQ.tex.8. a) Find the inverse, if it exists, of the matrix b) Verify your answer for part (a). c) Consider the matrix B = where is an unspecied parameter. For what values of (if any) does B not exist? MTH 9 SPRING 996 PRELIM # ; B ; C = Is invertible? If yes, explain why and nd. If no, explain why? MTH 9 SPRING 996 PRELIM # 6 9SP96PQ6.tex.8.9 Let I be the -by- identity matrix, let be the -by- matrix consisting of all zeros, let be the matrix in the question above, and let D be the 6-by-6 matrix given as four -by- blocks I Find D. D = I MTH 9 SPRING 996 FINL # 9 9SP96FQ9.tex.8. : Then the element in the rd row, nd column of is: a) - b) c) d) e) none of the above. C
7 .8. MTRIX INVERSE MTH 9 SPRING 996 FINL # 9SP96FQ.tex.8. Suppose What is the solution of x = b? a) b) c) d) e) none of the above ; b = MTH 9 FLL 996 PRELIM # 9F96PQ.tex.8.* Matrix algebra. Let [] be the matrix and let ~ b = : and let ~c = a) Find all solutions ~x to ~x = ~ b and check your answer by substitution. b) Find all solutions ~x to ~x = ~c and check your answer by substitution. c) Give a reason why you believe that does or does not exist. MTH 9 SPRING 99 PRELIM # 9SP9PQ.tex.8. Let M be the -by- matrix M = Find the inverse of M. C where
8 8 MTH 9 SPRING 99 FINL # 6 9SP9FQ6.tex.8. Find the inverse of matrix M. is a general matrix and I is the identity matrix. I M = I MTH 9 FLL 99 PRELIM # 9F9PQ.tex.8. Find the inverse of the 6 MTH 9 FLL 99 PRELIM # 9F9PQ.tex.8.6 Find the inverse of the I B I B I ; where each block is a by matrix, and I = I is the by identity matrix. Hint: the inverse is of the form = for certain E ; E ; I E E I E I ; MTH 9 SPRING 998 PRELIM # 9SP98PQ.tex.8. a) Find the inverse of the matrix : b) True or False? i) If and B are invertible, then B is the inverse of B. ii) If is an invertible n x n matrix, then the equation ~x = ~ b is consistent for each ~ b in < n. MTH 9 SPRING 998 PRELIM # 9SP98PQ.tex.8.8 Determine whether the following matrices are invertible. If they are invertible then nd the inverse. B =
9 .8. MTRIX INVERSE 9 MTH 9 FLL 998 PRELIM # bc 9F98PQbc.tex.8.9 (b) Is B = the inverse of? (c) Is the matrix if it does exist.) invertible? (You need not calculate the inverse MTH 9 SPRING?? PRELIM # 9SPxxPQ.tex.8. a) If and B are x matrices such that B = 6 ; show that the column space of is at least three dimensional. b) Find if 6 MTH 9 SPRING?? FINL # 9SPxxFQ.tex.8. a) Show 6 b) Find. c) Solve ~x = ~ b where ~ b = (; ; ) by using part (b). is nonsingular without nding. MTH 9 Unknown PRCTICE PRELIM? # 9UnknownPPQ.tex.8. a) For what value of a does the matrix B = not have an inverse? a b) n x n matrix C is said to be orthogonal if C t = C : Show that either det C = or det C = -. Hint: CC t = I: MTH 9 Unknown PRCTICE PRELIM # 9UnknownPPQ.tex.8. calculate the inverse of. Check your answer. Find the x matrix X if X = B where is the x matrix above and B =
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