- 5.7.1: For the matrix: a = 1 2 3 0 4 1 What is a12? What is a31?
- 5.7.2: Find x and y if c 1 3 x x + y d = c 1 3 2 6 d
- 5.7.3: Find x, y, z, and w if c x + y 2x 3y z w z + 2wd = c 4 7 6 6 d
- 5.7.4: If A is a symmetric matrix, find u, v, and w: a = 2 w u 7 0 v 1 3 4
- 5.7.5: Compute (if possible) a. a + D b. a D c. rB d. sC e. a + rD
- 5.7.6: Compute (if possible) a. b rC c. r(sC) e. D # C b. r(a + D) d. b # ...
- 5.7.7: Compute (if possible) a. a # C c. b # D b. C # a d. C2 = C # C
- 5.7.8: For a = 2 4 0 1 3 1 3 2 1 compute A3 = A # a # a.
- 5.7.9: For a = c 3 1 2 5 d b = c 4 1 2 1 d C = c 6 5 2 2 d compute (if pos...
- 5.7.10: If a = c 2 3 4 1 d b = c x 3 y 2 d find x and y if a # b = b # a
- 5.7.11: Prove that matrix multiplication is associative; that is, prove tha...
- 5.7.12: a. Prove that i 2 = i for any identity matrix I. b. Prove that i n ...
- 5.7.13: Let A and B be n n matrices. a. Prove that if A has one row consist...
- 5.7.14: An n n matrix A is diagonal if all elements aij with i j are 0. For...
- 5.7.15: The transpose of a matrix A, aT , is obtained by interchanging its ...
- 5.7.16: Prove that a # aT is symmetric for any matrix A (see Exercise 15)
- 5.7.17: Find two 2 2 matrices A and B such that a # b = 0 but a 0 and b 0.
- 5.7.18: Find three 2 2 matrices A, B, and C such that a # C = b # C, C 0, b...
- 5.7.19: If A and B are n n matrices, is it always true that (a + b) 2 = a2 ...
- 5.7.20: The vector of real numbers U = 3u1 u2 4 can be visualized on the re...
- 5.7.21: Prove that if a square matrix A is symmetric, then so is a2 , where...
- 5.7.22: Prove that if a square matrix A is symmetric, then so is a2n for an...
- 5.7.23: Let a = c 1 1 1 0 d For n 1, let F(n) equal the nth value in the Fi...
- 5.7.24: a. Show that for a = c 1 3 2 2 d b = c 12 34 12 14 d a # b = b # a ...
- 5.7.25: Prove that if A is invertible and r is a non-zero scalar, then rA i...
- 5.7.26: Prove that if A is invertible and a # b = a # C, then b = C.
- 5.7.27: For Exercises 2734, use Gaussian elimination to solve the systems o...
- 5.7.28: For Exercises 2734, use Gaussian elimination to solve the systems o...
- 5.7.29: For Exercises 2734, use Gaussian elimination to solve the systems o...
- 5.7.30: For Exercises 2734, use Gaussian elimination to solve the systems o...
- 5.7.31: For Exercises 2734, use Gaussian elimination to solve the systems o...
- 5.7.32: For Exercises 2734, use Gaussian elimination to solve the systems o...
- 5.7.33: For Exercises 2734, use Gaussian elimination to solve the systems o...
- 5.7.34: For Exercises 2734, use Gaussian elimination to solve the systems o...
- 5.7.35: Find an example of a system of 3 linear equations with 2 unknowns t...
- 5.7.36: Find an example of a system of 4 linear equations with 3 unknowns t...
- 5.7.37: You purchase an ancient Egyptian medallion at the State Fair from a...
- 5.7.38: Cell phone Plan A charges a flat monthly fee of $30.00 for the firs...
- 5.7.39: If A is an n n invertible matrix, the following method can be used ...
- 5.7.40: Use the method of Exercise 39 to find the inverse of matrix A in Ex...
- 5.7.41: Consider a system of n linear equations in n unknowns, such as the ...
- 5.7.42: In Exercises 4246, solve the systems of equations using the method ...
- 5.7.43: In Exercises 4246, solve the systems of equations using the method ...
- 5.7.44: In Exercises 4246, solve the systems of equations using the method ...
- 5.7.45: In Exercises 4246, solve the systems of equations using the method ...
- 5.7.46: In Exercises 4246, solve the systems of equations using the method ...
- 5.7.47: For Boolean matrices a = 1 0 0 1 1 0 0 1 1 b = 1 0 1 0 1 1 1 1 1 fi...
- 5.7.48: For Boolean matrices a = 0 0 1 1 1 0 1 0 0 b = 0 1 1 0 0 0 1 0 0 fi...
- 5.7.49: For Boolean matrices a = 0 1 0 1 0 1 0 0 1 b = 0 1 1 0 0 1 1 0 0 fi...
- 5.7.50: For Boolean matrices a = 1 1 0 0 1 1 0 0 1 b = 1 0 1 0 1 1 1 1 1 fi...
- 5.7.51: For Boolean matrices A and B, can it ever be the case that a ~ b = ...
- 5.7.52: For Boolean matrices A and B, prove that a ~ b = b ~ a and that a `...
- 5.7.53: How many distinct symmetric n n Boolean matrices are there?
- 5.7.54: Strassens algorithm reduces the amount of work to compute the produ...
Solutions for Chapter 5.7: Matrices
Full solutions for Mathematical Structures for Computer Science | 7th Edition
ISBN: 9781429215107
Solutions for Chapter 5.7
17
2
This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Since 54 problems in chapter 5.7: Matrices have been answered, more than 9759 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. Chapter 5.7: Matrices includes 54 full step-by-step solutions.
Key Math Terms and definitions covered in this textbook
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Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
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Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
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Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
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Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
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Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
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Toeplitz matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
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Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).