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Applied Linear Algebra 1st edition Peter J. Olver, Cheri Shakiban - Solutions

(a) Explain whyand are all legitimate permuted LU factorizations of the same matrix. List the elementary row operations that are being used in each case. (b) Use each of the factorizations to solve the linear system Do you always obtain the same result? Explain why or why not.
(a) Find three different permuted LU factorizations of the matrix(b) How many different permuted LU factorizations does A have?
What is the maximal number of permuted L U factorizations a regular 3×3 matrix can have? Give an example of such a matrix.
(a) Write a pseudocode program implementing the algorithm for finding the permuted LU factorization of a matrix.(b) Program your algorithm and test it on the examples in Exercise 1.4.19.
Solve the following systems of equations by Gaussian Elimination: (a) x1 - 2x2 + 2x3 = 15 x1 - 2x2 + x3 = 10 2x1 - x2 - 2x3 = -10 (b) 2x1 - x3 = 1 - 4x1 + 2x3 - 3x3 = -8 x1 - 3x2 + x3 = 5 (c) x2 - x3 = 4 -2x1 - 5x2 = 2 x1 + x3 = -8 (d) a - y + z - w = 0 -2a + 2y - z + u> = 2 -4a + 4y + 3z = 5
Find the equation z = ax + by + c for the plane passing through the three points p1 = (0, 2,-1), P2 = (-2,4,3), p3 = (2, -1, -3).
Show that a 2 Ã— 2 matrix(a) Nonsingular if and only if ad - bc ‰ 0. (b) Regular if and only if ad - bc ‰ 0 and Î± ‰ 0.
Explain why the solution to the homogeneous system Ax = 0 with nonsingular coefficient matrix is x = 0.
Write out the details of the proof of the "if" part of Theorem 1.7: if A is nonsingular, then the linear system Ax = b has a unique solution for every b. Theorem 1.7 A linear system Ax = b has a unique solution for every choice of right hand side b if and only if its coefficient matrix A is square
Write down the elementary 4 × 4 permutation matrix (a) P1 that permutes the second and fourth rows, and (b) P2 that permutes the first and fourth rows. (c) Do P1 and P2 commute? (d) Explain what the matrix products P1 P2 and P2 P1 do to a 4 × 4 matrix.
Verify by direct multiplication that the following matrices are inverses, i.e., both conditions in (1.36) hold:(a)(b) (c)
Explain how to write down the inverse permutation using the notation of Exercise 1.4.16. Apply your method to the examples in Exercise 1.5.9, and check the result by verifying that it produces the inverse permutation matrix.Notation of Exercise 1.4.16
Show thatis not invertible for any value of the entries.
Find all real 2×2 matrices that are their own inverses: A-1 = A.
Prove that if c ‰ 0 is any nonzero scalar and A is an invertible matrix, then the scalar product matrix c A is invertible, and
Prove that a diagonal matrix D = diag(d1,... , dn) is invertible if and only if all its diagonal entries are nonzero, in which case D-l = diag(1/d1..........1/dn) .
Prove that if U is a nonsingular upper triangular matrix, then the diagonal entries of U-1 are the reciprocals of the diagonal entries of U.
Two matrices A and B are said to be similar, written A ~ B, if there exists an invertible matrix S such that B = S-1 AS. Prove: (a) A ~ A (b) If A ~ B, then B ~ A. (c) If A ~ B ~ C, then A~ C.
(a) A block matrixIs called block diagonal if A and B are square matrices, not necessarily of the same size, while the O's are zero matrices of the appropriate sizes. Prove that D has an inverse if and only if both A and B do, and (b) Find the inverse of by using this method.
LetFind the right inverse of A by setting up and solving the linear system A X = I. Verify that the resulting matrix X is also a left inverse.
(a) Show thatIs a left in (b) Show that A does not have a right inverse. (c) Can you find any other left inverses of A?
Prove that the rectangular matrixhas a right inverse, but no left inverse.
(a) Are there any nonzero real scalars that satisfy (a + b)-l = a-l + B-l? (b) Are there any nonsingular real 2×2 matrices that satisfy (A + B)-1 = A-1 + B-1?
(a) Write down the elementary matrix that multiplies the third row of a 4 × 4 matrix by 7. (b) Write down its inverse.
Find the inverse of each of the following matrices, if possible, by applying the Gauss-Jordan Method.(a)(b) (c) (d) (e) (f) (g) (h) (i)
Write each of the matrices in Exercise 1.5.24 as a product of elementary matrices.Exercise 1.5.24(a)(b) (c) (d) (e) (f) (g) (h) (i)
Expressas a product of elementary matrices.
Use the Gauss-Jordan Method to find the inverse of the following complex matrices:(a)(b) (c) (d)
Can two nonsingular linear systems have the same solution and yet not be equivalent?
(b) Illustrate by adding -2 times the first row to the third row of
Write down the inverse of each of the following elementary matrices:(a)(b) (c) (d) (e) (f)
Solve the following systems of linear equations by computing the inverses of their coefficient matrices. (a) x + 2y = 1 x - 2y = -2 (b) 3u - 2v = 2 u+5v = l2 (c) x-y + 3z = 3 x - 2y+3z = -2 x -2y + z - 2 (d) y + 5z = 3 x - y + 3z = - 1 -2x + 3y = 5 (e) x + 4y - z = 3 2x+7y-2z = 5 -x - 5y + 2z = -7
For each of the nonsingular matrices in Exercise 1.5.24, use your computed inverse to solve the associated linear system A x = b, where b is the column vector of the appropriate size that has all 1's as its entries.Exercise 1.5.24Find the inverse of each of the following matrices, if possible, by
Produce the L D V or a permuted LDV factorization of the following matrices:(a)(b) (c) (d) (e) (f) (g)
Using the LDV factorization for the matrices you found in parts (a)-(g) of Exercise 1.5.32. solve the corresponding linear systems Ax = b, for the indicated vector b.(a)(b) (c) (d) (e) (f) (g)
Show that the inverse ofHowever, the inverse of What is M-1?
Explain why a matrix with a row of all zeros does not have an inverse.
(a) Write down the inverse of the matrices(b) Write down the product matrix C = AB and its inverse C-l using the inverse product formula.
(a) Find the inverse of the rotation matrixwhere Î¸ is a fixed angle. (b) Use your result to solve the system x = Î± cosÎ¸ - 6sinÎ¸, y = Î± sinÎ¸ + bcosÎ¸, for Î± and b in terms of x and y. (c) Prove
(a) Write down the inverses of each of the 3 × 3 permutation matrices (1.30). (b) Which ones are their own inverses, P-1 = P? (c) Can you find a non-elementary permutation matrix P which is its own inverse: P-1 = P?
Find the inverse of the following permutation matrices(a)(b) (c) (d)
Write down the transpose of the following matrices:(a)(b)(c)(d)(e) (1 2 -3)(f)(g)
If v, w are column vectors with the same number of entries, does vwT = wvT?
(a) Let A be an m Ã— n matrix. Let ei denote the 1 Ã— n column vector with a single 1 in the jth entry, as in (1.42). Explain why the product Aej equals the jth column of A.
Let A and B be m x n matrices. (a) Suppose that vTAw = vT B w for all vectors v, w. Prove that A = B. (b) Give an example of two matrices such that vT A v = vT B v for all vectors v, but A ≠ B.
(a) Explain why the inverse of a permutation matrix equals its transpose: P-1 = PT. (b) If A-1 = AT, is A necessarily a permutation matrix?
Let A be a square matrix and P a permutation matrix of the same size. (a) Explain why the product APT has the effect of applying the permutation defined by P to the columns of A. (b) Explain the effect of multiplying PAPT.
Let v, w be n Ã— 1 column vectors.(a) Prove that in most cases the inverse of the /; x n matrix A = I - vwT has the form A-1 = I- cvwT for some scalar c. Find all v, w for which such a result is valid.(b) Illustrate the method when(c) What happens when the method fails?
Find all values of a. b. and c for which the following matrices are symmetric:(a)(b) (c)
List all symmetric (a) 3×3 permutation matrices, (b) 4 × 4 permutation matrices.
True or false: If A is symmetric, then A2 is symmetric.
(a) Show that every diagonal matrix is symmetric. (b) Show that an upper (lower) triangular matrix is symmetric if and only if it is diagonal.
Let A be a symmetric matrix. (a) Show that An is symmetric for any nonnegative integer n. (b) Show that 2A2 - 3A + I is symmetric. (c) Show that any matrix polynomial p(A) of A, cf. Exercise 1.2.35, is a symmetric matrix.
Show that if A is any matrix, then K = ATA and L = A AT are both well-defined, symmetric matrices.
Find the LDLT factorization of the following sym- metric matrices:(a)(b) (c) (d)
Find the LDLT factorization of the matricesand
Prove that the 3 Ã— 3 matrixcannot be factored as A= LDLT.
Suppose A = L U is a regular matrix. Write down the L U factorization of AT. Prove that AT is also regular, and its pivots are the same as the pivots of A.
Skew-Symmetric Matrices: An n × n matrix J is called skew-symmetric if JT = - J. (a) Show that every diagonal entry of a skew- symmetric matrix is zero. (b) Write down an example of a nonsingular skew- symmetric matrix. (c) Can you find a regular skew-symmetric matrix? (d) Show that if J is a
Show that (A B)T = ATBT if and only if A and B are square commuting matrices.
(a) Prove that every square matrix can be expressed as the sum, A = S + J, of a symmetric matrix S = ST and a skew-symmetric matrix J = -JT.(b) Writeas the sum of symmetric and skew-symmetric matrices.
Prove formula (1.53). (AB)T = BTAT. (1.53)
A square matrix is called normal if it commutes with its transpose: ATA = AAT. Find all normal 2×2 matrices.
(a) Prove that the inverse transpose operation (1.55) respects matrix multiplication: (AB)-T = A-TB-T.(b) Verify this identity for
Solve the following linear systems by (i) Gaussian Elimination with Back Substitution; (ii) The Gauss-Jordan algorithm to convert the augmented matrix to the fully reduced form ( 1 | x) with solution x; (iii) Computing the inverse of the coefficient matrix, and then multiplying it with the right
(a) Find the L U factorization of the n × n tridiagonal matrix An with all 2's along the diagonal and all - 1's along the sub- and super-diagonals for n = 3, 4 and 5. (b) Use your factorizations to solve the system Anx = b, where b=(l,l,l,...,l)T. (c) Do the entries in the factors approach a limit
Answer Exercise 1.7.10 if the super-diagonal entries of An are changed to +1. Exercise 1.7.10 (a) Find the L U factorization of the n × n tridiagonal matrix An with all 2's along the diagonal and all - 1's along the sub- and super-diagonals for n = 3, 4 and 5. (b) Use your factorizations to solve
Find the L U factorizations ofDo you see a pattern? Try the 6 Ã— 6 version. The following exercise should now be clear.
A tricirculant matrixis tridiagonal except for its (1. n) and (u. 1) entries. Tricirculant matrices arise in the numerical solution of periodic boundary value problems and in spline interpolation. (a) Prove that if C = LU is regular, its factors have the form (b) Compute the LU factorization of the
A matrix A is said to have band width k if all entries that are more than k slots away from the main diagonal are zero: aij = 0 whenever |i - j| > k. (a) Show that a tridiagonal matrix has band width1. (b) Write down an example of a 6 × 6 matrix of band width 2 and one of band width 3. (c) Prove
(a) Find the exact solution to the linear system(b) Solve the system using Gaussian Elimination with 2 digit rounding. (c) Solve the system using Partial Pivoting and 2 digit rounding. (d) Compare your answers and discuss.
(a) Find the exact solution to the linear system x - 5 y - z = l.1/6x - 5/6y + z = 0,2x-y = 3. (b) Solve the system using Gaussian Elimination with 4 digit rounding. (c) Solve the system using Partial Pivoting and 4 digit rounding. Compare your answers.
Answer Exercise 1.7.17 for the system x + 4y - 3z = -3. 25x + 97y - 35z = 39. 35x - 22y + 33 z = -15. Exercise 1.7.17 (a) Find the exact solution to the linear system x - 5 y - z = l.1/6x - 5/6y + z = 0,2x-y = 3. (b) Solve the system using Gaussian Elimination with 4 digit rounding. (c) Solve the
Employ 2 digit arithmetic with rounding to compute an approximate solution of the linear system 0.2x + 2y - 3z = 6, 5x + 43y + 27z = 58, 3x + 23.v - 42z = -87. Using the following methods: (a) Regular Gaussian Elimination with Back Substitution; (b) Gaussian Elimination with Partial Pivoting: (c)
(a) Let A be an n × n matrix. Which is faster to compute, A1 or A-1? Justify your answer. (b) What about A3 versus A-1? (c) How many operations are needed to compute Ak?
Solve the following systems by hand, using pointers instead of physically interchanging the rows:(a)(b) (c) (d)
Solve the following systems using Partial Pivoting and pointers:(a)(b) (c) (d)
(a) Write out a pseudo-code algorithm, using both row and column pointers, for Gaussian Elimination with Full Pivoting. (b) Implement your code on a computer, and try it on the systems in Exercise 1.7.21.
Implement the computer experiment with Hilbert matrices outlined in the last paragraph of the section.
Let Hn be the n Ã— n Hilbert matrix (1.70), and Kn = H-ln its inverse. It can be proved, [32, p. 513], that the (i, j) entry of Kn isWhere is the standard binomial coefficient. (a) Write down the inverse of the Hilbert matrices H3, H4, H5 by either using the formula or using the
Use induction to prove the summation formulae (1.60). (1.61) and (1.62).
Let A be a general n × n matrix. Determine the exact number of arithmetic operations needed to compute A-1 using (a) Gaussian Elimination to factor P A = LU and then Forward and Back Substitution to solve the n linear systems (1.63); (b) The Gauss-Jordan method. Make sure your totals do not count
Count the number of arithmetic operations needed to solve a system the "old-fashioned" way, by using elementary row operations of all three types, in the same order as the Gauss-Jordan scheme, to fully reduce the augmented matrix M = (A | b) to the form (I | d), with x = d being the solution.
Here, we describe a remarkable algorithm for matrix multiplication discovered by Strassen, [62].Letand be block matrices of size n = 2m, where all blocks are of size m Ã— m. (a) Let D1 = (A1 + A4) (B1 + B4), D2 = (A1 - A3) (B1 + B2). D3 = (A2 - A4) (B3 + B4), D4 = (A1 + A2) Bi, D5 =
For each of the following tridiagonal systems find the L U factorization of the coefficient matrix, and then solve the system.(a)(b) (c)
Which of the following systems has (i) A unique solution? (ii) Infinitely many solutions? (iii) No solution? In each case, find all solutions: (a) x-2y = 1 3x + 2y = -3 (b) 2x + y + 3z = l x + 4y - 2z = -3 (c) x + y-2z = -3 2x-y + 3z = 7 x - 2y + 5z = 1 (d) x-2y + z = 6 2x
Find a coefficient matrix A such that the associated linear system Ax = b has (a) Infinitely many solutions for every b, (b) 0 or ∞ solutions, depending on b, (c) 0 or 1 solution depending on b. (d) Exactly 1 solution for all b.
Give an example of a nonlinear system of two equations in two unknowns that has (a) No solution (b) Exactly two solutions, (c) Exactly three solutions, (d) Infinitely many solutions.
(a) Prove that the product A = vwT of a nonzero m Ã— 1 column vector v by a nonzero 1 Ã— n row vector wT is an m Ã— n matrix of rank r = 1.(b) Compute the following rank one products:(i)(ii) (iii) (c) Prove that every rank one matrix can be written in the form A
Find two matrices A.B such that rank AB ≠ rank BA.
Let A be an m x n matrix of rank r.(a) Suppose C = (A B) is an m Ã— k matrix, k > n, whose first n columns are the same as the columns of A. Prove that rank C ‰¥ rank A. Give an example with rank C = rank A; with rankC > rank A.(b) LetBe a j Ã— n matrix, j > m,
Determine if the following systems are compatible and, if so, find the general solution: (a) 6x1 +3x2 = 12 4x1 + 2x2 = 9 (b) 8x1 + 12x2 = 16 6x1 + 9x2 = 13 (c) x1 + 2x2 = 1 2x1 + 5x2 = 2 3x1 + 6x2 = 3 (d) 2x1 - 6x2 + 4x3 = 2 - xi + 3x2 - 2x3 = -1 (e) 2x1 + 2x2 + 3x3 = 1 x2 + 2x3 = 3 4x1 + 5x2 +
Let A be a singular square matrix. Prove that there exist elementary matrices E1,... . EN such that A = E1 E2 ∙ ∙ ∙ En Z. where Z is a matrix with at least one all zero row.
(a) If A is an m x n matrix and M = (A | b) the augmented matrix for the linear system Ax = b. Show that either (i) Rank A = rank M, or (ii) Rank A = rank M - 1. (b) Prove that the system is compatible if and only if case (i) holds.
Solve the following homogeneous linear systems. (a) x + y - 2z = 0 -x + 4y -3z = 0 (b) 2x + 3y - z = 0 -4x+3y-5z = 0 x - 3y + 3z = 0 (c) -x+y-4z = 0 - 2x + 2y - 6z = 0 x + 3y + 3z = 0 (d) x +2y-2z + w = 0 -3x + z - 2w = 0 (e) -x+3y - 2z+w=0 -2x + 5y + z-2w =0 3x-8y-z - Aw = 0 (f) -y + z = 0 2x
Find all solutions to the homogeneous system Ax = 0 for the coefficient matrix(a)(b) (c) (d) (e) (f) (g) (h)
Let U be an upper triangular matrix. Show that the homogeneous system U x = 0 admits a nontrivial solution if and only if U has at least one 0 on its diagonal.
Find the solution to the homogeneous system 2x1 + x2 - 2x3 = 0. 2x1 - x2 - 2x3 = 0. Then solve the inhomogeneous version where the right hand sides are changed to a, h, respectively. What do you observe?\
Answer Exercise 1.8.25 for the system 2x1 + x2 + x3 - x4 = 0, 2x1 - 2x2 -x3 + 3x4 = 0.
Find all values of k for which the following homogeneous systems of linear equations have a nontrivial solution: (a) x + k y = 0 kx + 4y = 0 (b) x1 + kx2 + 4x3 = 0 kx1 + X2 + 2x3 = 0 2x1 + kx2 +8x3 = 0 (c) x + ky+2z = 0 3x -ky-2z = 0 (k + l)x - 2y - 4z = 0 kx + 3y + 6z = 0

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