 3.4.1: A rectangular array of numbers, arranged in rows and columns and pl...
 3.4.2: The augmented matrix for the system e 3x  2y = 6 4x + 5y = 8 is ...
 3.4.3: The augmented matrix for the system 2x + y + 4z = 4 3x + z = 1 4x ...
 3.4.4: Consider the matrix c 2 1 5 4 2 ` 7 d . We can obtain 1 in the sh...
 3.4.5: Consider the matrix 1 1 1 2 3 4 2 4 2 1 1 6 . We can obtain 0 i...
 3.4.6: True or false: Two columns of a matrix may be interchanged to form ...
 3.4.7: True or false: The matrix c 1 2 ` 5 0 0 8 d represents an inconsist...
 3.4.8: J 1  2 5 4 2 2 3 4 1 R 4R1 + R2
 3.4.9: C 2 6 4 1 5 5 3 0 4 3 10 0 7 S 1 2 R1
 3.4.10: C 3 12 6 1 4 4 2 0 7 3 9 0 4 S 1 3 R1
 3.4.11: C 1 3 2 3 1 1 2 2 1 3 0 7 3 S 3R1 + R2
 3.4.12: C 1 1 5 3 3 1 1 3 2 3 6 10 5 S 3R1 + R2
 3.4.13: C 1 1 1 2 1 1 3 1 1 3 6 3 4 S 2R1 + R2 3R1 + R3
 3.4.14: C 1 2 1 2 1 2 1 3 2 3 2 5 8 S 2R1 + R2 1R1 + R3
 3.4.15: e x + y = 6 x  y = 2
 3.4.16: e x + 2y = 11 x  y = 1
 3.4.17: e 2x + y = 3 x  3y = 12
 3.4.18: e 3x  5y = 7 x  y = 1
 3.4.19: e 5x + 7y = 25 11x + 6y = 8
 3.4.20: e 3x  5y = 22 4x  2y = 20
 3.4.21: e 4x  2y = 5 2x + y = 6
 3.4.22: e 3x + 4y = 12 6x  8y = 16
 3.4.23: e x  2y = 1 2x + 4y = 2
 3.4.24: 3x  6y = 1 2x  4y = 2 3
 3.4.25: x + y  z = 2 2x  y + z = 5 x + 2y + 2z = 1
 3.4.26: x  2y  z = 2 2x  y + z = 4 x + y  2z = 4
 3.4.27: x + 3y = 0 x + y + z = 1 3x  y  z = 11
 3.4.28: 3y  z = 1 x + 5y  z = 4 3x + 6y + 2z = 11
 3.4.29: 2x + 2y + 7z = 1 2x + y + 2z = 2 4x + 6y + z = 15
 3.4.30: 3x + 2y + 3z = 3 4x  5y + 7z = 1 2x + 3y  2z = 6
 3.4.31: x + y + z = 6 x  z = 2 y + 3z = 11
 3.4.32: x + y + z = 3 y + 2z = 1 x + z = 0
 3.4.33: x  y + 3z = 4 2x  2y + 6z = 7 3x  y + 5z = 14
 3.4.34: 3x  y + 2z = 4 6x + 2y  4z = 1 5x  3y + 8z = 0
 3.4.35: x  2y + z = 4 5x  10y + 5y = 20 2x + 4y  2z = 8
 3.4.36: x  3y + z = 2 4x  12y + 4z = 8 2x + 6y  2z = 4
 3.4.37: x + y = 1 y + 2z = 2 2x  z = 0
 3.4.38: x + 3y = 3 y + 2z = 8 x  z = 7
 3.4.39: D 1 1 1 1 0 1 2 1 0 0 1 6 0 0 0 1 4 3 0 17 3 T
 3.4.40: D 1 2 1 0 0 1 1 2 0 0 1 1 0 0 0 1 4 2 3 2 3 T
 3.4.41: D 1 1 1 1 0 1 2 1 2 0 3 4 5 1 2 4 4 3 0 11 6 T 2R1 + R3 5R1 + R4
 3.4.42: D 1 5 2 2 0 1 3 1 3 0 2 1 4 1 4 2 4 4 0 6 3 T 3R1 + R3 4R1 ...
 3.4.43: w + x + y + z = 4 2w + x  2y  z = 0 w  2x  y  2z = 2 3w + 2x ...
 3.4.44: w + x + y + z = 5 w + 2x  y  2z = 1 w  3x  3y  z = 1 2w  x ...
 3.4.45: A ball is thrown straight upward. The graph shows the balls height,...
 3.4.46: A football is kicked straight upward. The graph shows the footballs...
 3.4.47: For single women in the poll, the percentage who said no exceeded t...
 3.4.48: For single men in the poll, the percentage who said no exceeded the...
 3.4.49: What is a matrix?
 3.4.50: Describe what is meant by the augmented matrix of a system of linea...
 3.4.51: In your own words, describe each of the three matrix row operations...
 3.4.52: Describe how to use matrices and row operations to solve a system o...
 3.4.53: When solving a system using matrices, how do you know if the system...
 3.4.54: When solving a system using matrices, how do you know if the system...
 3.4.55: Most graphing utilities can perform row operations on matrices. Con...
 3.4.56: If your graphing utility has a REF (rowechelon form) command, use ...
 3.4.57: A matrix with 1s down the main diagonal and 0s in every position ab...
 3.4.58: Matrix row operations remind me of what I did when solving a linear...
 3.4.59: When I use matrices to solve linear systems, the only arithmetic in...
 3.4.60: When I use matrices to solve linear systems, I spend most of my tim...
 3.4.61: Using row operations on an augmented matrix, I obtain a row in whic...
 3.4.62: A matrix row operation such as  4 5R1 + R2 is not permitted becaus...
 3.4.63: The augmented matrix for the system x  3y = 5 y  2z = 7 2x + z = ...
 3.4.64: In solving a linear system of three equations in three variables, w...
 3.4.65: The row operation kRi + Rj indicates that it is the elements in row...
 3.4.66: The vitamin content per ounce for three foods is given in the follo...
 3.4.67: If f(x) = 3x + 10, find f(2a  1). (Section 2.1, Example3)
 3.4.68: If f(x) = 3x and g(x) = 2x  3, find (fg)(1). (Section 2.3, Exampl...
 3.4.69: Simplify: 4x8 y12 12x3 y24 . (Section 1.6, Example 9)
 3.4.70: 2(5)  (3)(4)
 3.4.71: 2(5)  1(4) 5(5)  6(4)
 3.4.72: 2(30  (3))  3(6  9) + (1)(1  15)
Solutions for Chapter 3.4: Matrix Solutions of Linear Systems
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 3.4: Matrix Solutions of Linear Systems
Get Full SolutionsChapter 3.4: Matrix Solutions of Linear Systems includes 72 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. Since 72 problems in chapter 3.4: Matrix Solutions of Linear Systems have been answered, more than 19849 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Column space C (A) =
space of all combinations of the columns of A.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·

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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.