 10.1.1: A rectangular array of real numbers that can be used to solve a sys...
 10.1.2: A matrix is ________ if the number of rows equals the number of col...
 10.1.3: For a square matrix, the entries are the ________ ________ entries
 10.1.4: A matrix with only one row is called a ________ matrix, and a matri...
 10.1.5: The matrix derived from a system of linear equations is called the ...
 10.1.6: The matrix derived from the coefficients of a system of linear equa...
 10.1.7: Two matrices are called ________ if one of the matrices can be obta...
 10.1.8: A matrix in rowechelon form is in ________ ________ ________ if ev...
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 10.1.14: 7 0 6 5 4 1 33
 10.1.15: 4x x 3y 3y 5 12
 10.1.16: 7x 4y 22 5x 9y 15
 10.1.17: x 10y 5x 3y 2x y 2z 2 4z 0
 10.1.18: x 7x 3x 8y y 5z 15z 8z 8 38 20 x
 10.1.19: 7x 19x 5y z 13 8z 10
 10.1.20: 9x 2y 3z 25y 11z 20 5
 10.1.21: 1 2 2 3 7 4 w
 10.1.22: 7 8 5 3 0 2
 10.1.23: 2 0 6 0 1 3 5 2 0 12 7 2 7
 10.1.24: 4 11 3 5 0 8 1 6 0 18 25 29 2
 10.1.25: 9 2 1 3 12 18 7 0 3 5 8 2 0 2 0 0 0 10 4 10 4 11
 10.1.26: 6 1 4 0 2 0 1 8 1 7 10 1 5 3 6 11 25 7 23 21 9 2 1
 10.1.27: 1 2 4 10 3 5
 10.1.28: 3 4 6 3 8 6
 10.1.29: 1 5 1 2 1 4
 10.1.30: 3 18 3 8 12 4 1
 10.1.31: 0 0 5 1 0 4 2 1 1 2 7
 10.1.32: 0 0 0 1 0 6 0 1 1 7 3
 10.1.33: 1 0 0 1 1 3 4 2 5 1 6 5
 10.1.34: 1 0 0 2 2 4 7 3 2 1 2 1 0
 10.1.35: 3 3 0 1 39 8 2 3
 10.1.36: 3 5 1 0 4 5 3 4
 10.1.37: 1 0 0 3 1 7 7 5 27 6 5 27 0 1
 10.1.38: 2 5 2 5 4 3 1 7 2 7 6 1
 10.1.39: Perform the sequence of row operations on the matrix. What did the ...
 10.1.40: Perform the sequence of row operations on the matrix. What did the ...
 10.1.41: 1 0 0 0 1 0 0 1 0 0 5 0
 10.1.42: 1 0 0 3 0 0 0 1 0 0 8 0
 10.1.43: 1 0 0 0 1 0 0 0 0 1 1 2
 10.1.44: 1 0 0 0 1 0 1 0 1 0 2 0
 10.1.45: 1 2 3 1 1 6 0 2 7 5 10 14
 10.1.46: 1 3 2 2 7 1 1 5 3 3 14 8 1
 10.1.47: 1 5 6 1 4 8 1 1 18 1 8 0
 10.1.48: 1 3 4 3 10 10 0 1 2 7 23 24 1
 10.1.49: 3 1 2 3 0 4 3 4 2
 10.1.50: 1 5 2 3 15 6 2 9 10
 10.1.51: 1 1 2 4 2 2 4 8 3 4 4 11 5 9 3 14 1
 10.1.52: 2 4 1 3 3 2 5 8 1 5 2 10 2 8 0 30 1
 10.1.53: 3 1 5 1 1 1 12 4
 10.1.54: 5 1 1 5 2 10 4 32
 10.1.55: 1 0 2 1 4 3 x,
 10.1.56: 1 0 5 1 0 1
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 10.1.60: 1 0 0 1 6 10
 10.1.61: 1 0 0 0 1 0 0 0 1 4 10 4
 10.1.62: 1 0 0 0 1 0 0 0 1 5 3 0
 10.1.63: x 2y 7 2x y 8
 10.1.64: 2x 6y 16 2x 3y 7
 10.1.65: 3x 2y x 3y 27 13
 10.1.66: x y 2x 4y 4 34 3
 10.1.67: 2x 6y x 2y 22 9
 10.1.68: 5x 5y 2x 3y 5 7 2
 10.1.69: 8x 4y 5x 2y 7 1
 10.1.70: x 3y 2x 6y 5 10
 10.1.71: x 3x 2x y 2y 3z 2z z 2 5 4
 10.1.72: 2x y 2y 7x 5y 3z 24 z 14 6 x
 10.1.73: x 2x 3x y y 2y z z z 14 21 19
 10.1.74: 2x x x 2y 3y y z z 2 28 14 x
 10.1.75: x x 2y 4y y 3z 2z z 28 0 5 2
 10.1.76: 3x x x 2y y y z 2z 4z 15 10 14 x
 10.1.77: 0 x x 2y y 0 0
 10.1.78: x 2x 2y 4y 0 0
 10.1.79: x 3x 2y 7y z 6z 8 26
 10.1.80: x 2x y y 4z z 5 9
 10.1.81: x y 3x 4y 4x 8y 22 4 32
 10.1.82: x 2y 0 x y 6 3x 2y 8
 10.1.83: 3x x 2x x 2y y y y z 4z 2z z w 2w w w 0 25 2 6
 10.1.84: x 3x 4x 2x 4y 2y 3y y 3z z 2z 4z 2w 4w w 3w 9 13 4 10 3x x
 10.1.85: 3x 3y 12z 6 x y 4z 2 2x 5y 20z 10 x 2y 8z 4
 10.1.86: 2x 10y 2z x 5y 2z x 5y z 3x 15y 3z 6 6 3 9 3
 10.1.87: 2x y 3x 4y x 5y 5x 2y z 2w w 2z 6w z w 6 1 3 3
 10.1.88: x 2y 2z 4w 3x 6y 5z 12w x 3y 3z 2w 6x y z w 11 30 5 9 2
 10.1.89: x 2x 3x y 3y 5y z z z w 0 2w 0 0
 10.1.90: x 2y x y y z 3w 0 w 0 z 2w 0
 10.1.91: (a) (b)
 10.1.92: (a) (b)
 10.1.93: (a) (b)
 10.1.94: (a) (b)
 10.1.95: f 1 1, f 2 1, f 3 5
 10.1.96: f 1 2, f 2 9, f 3 20
 10.1.97: f 2 15, f 1 7, f 1 3 Hor
 10.1.98: f 2 3, f 1 3, f 2 11 f 2
 10.1.99: f 1 5 f
 10.1.100: f 1 4
 10.1.101: f 2 7 f
 10.1.102: f 2 17 f
 10.1.103: Use the system to write two different matrices in rowechelon form ...
 10.1.104: ELECTRICAL NETWORK The currents in an electrical network are given ...
 10.1.105: PARTIAL FRACTIONS Use a system of equations to write the partial fr...
 10.1.106: PARTIAL FRACTIONS Use a system of equations to write the partial fr...
 10.1.107: FINANCE A small shoe corporation borrowed $1,500,000 to expand its ...
 10.1.108: FINANCE A small software corporation borrowed $500,000 to expand it...
 10.1.109: TIPS A food server examines the amount of money earned in tips afte...
 10.1.110: BANKING A bank teller is counting the total amount of money in a ca...
 10.1.111: (1, 8) (2, 13) (3, 20) x y
 10.1.112: (3, 5) (1, 9) x 8 y
 10.1.113: MATHEMATICAL MODELING A video of the path of a ball thrown by a bas...
 10.1.114: DATA ANALYSIS: SNOWBOARDERS The table shows the numbers of people (...
 10.1.115: Water flowing through a network of pipes (in thousands of cubic met...
 10.1.116: The flow of traffic (in vehicles per hour) through a network of str...
 10.1.117: 5 1 0 3 2 6 7 0 x
 10.1.118: The method of Gaussian elimination reduces a matrix until a reduced...
 10.1.119: THINK ABOUT IT The augmented matrix below represents system of line...
 10.1.120: THINK ABOUT IT (a) Describe the rowechelon form of an augmented ma...
 10.1.121: Describe the three elementary row operations that can be performed ...
 10.1.122: CAPSTONE I
 10.1.123: What is the relationship between the three elementary row operation...
Solutions for Chapter 10.1: Matrices and Systems of Equations
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 10.1: Matrices and Systems of Equations
Get Full SolutionsChapter 10.1: Matrices and Systems of Equations includes 123 full stepbystep solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Since 123 problems in chapter 10.1: Matrices and Systems of Equations have been answered, more than 51089 students have viewed full stepbystep solutions from this chapter.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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·