 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 nonzero 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: Matrices
Get Full SolutionsThis 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 4196 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Mathematical Structures for Computer Science was written by Patricia and is associated to the ISBN: 9781429215107. Chapter 5.7: Matrices includes 54 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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 •

Outer product uv T
= column times row = rank one matrix.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.
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