 7.3.7.1.119: A rectangular array of numbers is called a(n) .
 7.3.7.1.120: A matrix that contains the same number of rows and columns is calle...
 7.3.7.1.121: The number of columns in a 3 * 2 matrix is .
 7.3.7.1.122: The number of rows in a 4 * 3 matrix is .
 7.3.7.1.123: To add or subtract two matrices, the matrices must have the same .
 7.3.7.1.124: When multiplying a matrix by a real number, the real number is call...
 7.3.7.1.125: Multiplication of matrices is possible only when the number of of t...
 7.3.7.1.126: The product of a square matrix A with an identity matrix I is alway...
 7.3.7.1.127: The multiplicative identity matrix of a 2 * 2 matrix is .
 7.3.7.1.128: The multiplicative identity matrix of a 3 * 3 matrix is .
 7.3.7.1.129: In Exercises 1114, determine A + B.A = c 8 5 7 1 d , B = c 3 2 0 4 d
 7.3.7.1.130: In Exercises 1114, determine A + B.A = c 7 4 1 5 3 6 d , B = c 1...
 7.3.7.1.131: In Exercises 1114, determine A + B.A = C 5 2 1 4 7 0 S , B = C 3 ...
 7.3.7.1.132: In Exercises 1114, determine A + B.A = C 2 63 1 6 4 3 05 S , B = ...
 7.3.7.1.133: In Exercises 1518, determine A  B.A = c 8 6 4 2 d , B = c 2 5 9 1 d
 7.3.7.1.134: In Exercises 1518, determine A  B.A = C 10 1 12 2 3 9 S , B = C ...
 7.3.7.1.135: In Exercises 1518, determine A  B.A = C 5 1 8 6 1 5 S , B = C 6...
 7.3.7.1.136: In Exercises 1518, determine A  B.A = C 5 3 1 742 6 1 5 S , B =...
 7.3.7.1.137: In Exercises 1924, A = c 1 2 0 5 d , B = c 3 2 5 0 d , and C = c 2...
 7.3.7.1.138: In Exercises 1924, A = c 1 2 0 5 d , B = c 3 2 5 0 d , and C = c 2...
 7.3.7.1.139: In Exercises 1924, A = c 1 2 0 5 d , B = c 3 2 5 0 d , and C = c 2...
 7.3.7.1.140: In Exercises 1924, A = c 1 2 0 5 d , B = c 3 2 5 0 d , and C = c 2...
 7.3.7.1.141: In Exercises 1924, A = c 1 2 0 5 d , B = c 3 2 5 0 d , and C = c 2...
 7.3.7.1.142: In Exercises 1924, A = c 1 2 0 5 d , B = c 3 2 5 0 d , and C = c 2...
 7.3.7.1.143: In Exercises 2530, determine A * B.A = c 3 5 0 6 d , B = c 4 2 8 3 d
 7.3.7.1.144: In Exercises 2530, determine A * B.A = c 1 1 2 6 d , B = c 4 2 3...
 7.3.7.1.145: In Exercises 2530, determine A * B.A = c 2 3 1 04 6 d , B = C 2 4 1 S
 7.3.7.1.146: In Exercises 2530, determine A * B.A = c 1 1 1 1 d , B = c 1 1 1 2 d
 7.3.7.1.147: In Exercises 2530, determine A * B.A = C 476 231 512 S , B = C 100...
 7.3.7.1.148: In Exercises 2530, determine A * B.A = c 2 3 9 2 d , B = c 1 2 3 4 d
 7.3.7.1.149: In Exercises 3136, determine A + B and A * B. If an operation canno...
 7.3.7.1.150: In Exercises 3136, determine A + B and A * B. If an operation canno...
 7.3.7.1.151: In Exercises 3136, determine A + B and A * B. If an operation canno...
 7.3.7.1.152: In Exercises 3136, determine A + B and A * B. If an operation canno...
 7.3.7.1.153: In Exercises 3136, determine A + B and A * B. If an operation canno...
 7.3.7.1.154: In Exercises 3136, determine A + B and A * B. If an operation canno...
 7.3.7.1.155: In Exercises 3739, show the commutative property of addition, A + B...
 7.3.7.1.156: In Exercises 3739, show the commutative property of addition, A + B...
 7.3.7.1.157: In Exercises 3739, show the commutative property of addition, A + B...
 7.3.7.1.158: Create two matrices with the same dimensions, A and B, and show tha...
 7.3.7.1.159: In Exercises 41 43, show that the associative property of addition,...
 7.3.7.1.160: In Exercises 41 43, show that the associative property of addition,...
 7.3.7.1.161: In Exercises 41 43, show that the associative property of addition,...
 7.3.7.1.162: Create three matrices with the same dimensions, A, B, and C, and sh...
 7.3.7.1.163: In Exercises 45 49, determine whether the commutative property of m...
 7.3.7.1.164: In Exercises 45 49, determine whether the commutative property of m...
 7.3.7.1.165: In Exercises 45 49, determine whether the commutative property of m...
 7.3.7.1.166: In Exercises 45 49, determine whether the commutative property of m...
 7.3.7.1.167: In Exercises 45 49, determine whether the commutative property of m...
 7.3.7.1.168: Create two square matrices A and B with the same dimensions, and de...
 7.3.7.1.169: In Exercises 5155, show that the associative property of multiplica...
 7.3.7.1.170: In Exercises 5155, show that the associative property of multiplica...
 7.3.7.1.171: In Exercises 5155, show that the associative property of multiplica...
 7.3.7.1.172: In Exercises 5155, show that the associative property of multiplica...
 7.3.7.1.173: In Exercises 5155, show that the associative property of multiplica...
 7.3.7.1.174: Create three matrices, A, B, and C, and show that (A * B) * C = A *...
 7.3.7.1.175: MODELINGHigh School Play Bakersfield High School sold tickets for t...
 7.3.7.1.176: MODELINGTshirt Inventory Hollister sells two types of tshirtsmale...
 7.3.7.1.177: MODELINGSupplying Muffins A bakery supplies chocolate chip, blueber...
 7.3.7.1.178: MODELINGMenu Choices The cafeteria manager at the University of Vir...
 7.3.7.1.179: MODELINGCookie Company Costs The Original Cookie Factory bakes and ...
 7.3.7.1.180: In Exercises 62 and 63, use the information given in Exercise 61. S...
 7.3.7.1.181: In Exercises 62 and 63, use the information given in Exercise 61. S...
 7.3.7.1.182: MODELINGFood Prices To raise money for a local charity, the Spanish...
 7.3.7.1.183: In Exercises 65 and 66, there are many acceptable answers.a) Constr...
 7.3.7.1.184: In Exercises 65 and 66, there are many acceptable answers.a) Constr...
 7.3.7.1.185: Two matrices whose product is the multiplicative identity matrix ar...
 7.3.7.1.186: Two matrices whose product is the multiplicative identity matrix ar...
 7.3.7.1.187: In Exercises 69 and 70, determine whether the statement is true or ...
 7.3.7.1.188: In Exercises 69 and 70, determine whether the statement is true or ...
 7.3.7.1.189: MODELINGSofa Manufacturing Costs The number of hours of labor requi...
 7.3.7.1.190: Is it possible that two matrices could be added but not multiplied?...
 7.3.7.1.191: Is it possible that two matrices could be multiplied but not added?...
 7.3.7.1.192: Make up two matrices A and B such that A + B = c 1 0 0 1 d and A * ...
 7.3.7.1.193: Find an article that shows information illustrated in matrix form. ...
 7.3.7.1.194: Messages The study of encoding and decoding messages is called cryp...
Solutions for Chapter 7.3: Systems of Linear Equations and Inequalities
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 7.3: Systems of Linear Equations and Inequalities
Get Full SolutionsThis textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Chapter 7.3: Systems of Linear Equations and Inequalities includes 76 full stepbystep solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. Since 76 problems in chapter 7.3: Systems of Linear Equations and Inequalities have been answered, more than 79807 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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].

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

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).

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.