 52.1: Find the value of the variable:
 52.2: Find the value of the variable:
 52.3: Find the value of the variable:
 52.4: Find the value of the variable:
 52.5: Find the value of the variable:
 52.6: Find the value of the variable:
 52.7: Find the value of the variable:
 52.8: Find the value of the variable:
 52.9: Find the value of the variable:
 52.10: Find the value of the variable:
 52.11: The difference in hours between fulltimers and the parttimers who ...
 52.12: Manny plans to save of his salary each week. If his weekly salary i...
 52.13: Last week at the Sunshine Valley Rock Festival, Joel sold 3 times a...
 52.14: Elaine sold 3 times as many magazine subscriptions as Ron did. Ron ...
 52.15: Will ordered 2 times as many boxes of ballpoint pens as boxes of fe...
 52.16: A real estate salesperson bought promotional calendars and date boo...
Solutions for Chapter 52: USING EQUATIONS TO SOLVE PROBLEMS
Full solutions for Business Math,  9th Edition
ISBN: 9780135108178
Solutions for Chapter 52: USING EQUATIONS TO SOLVE PROBLEMS
Get Full SolutionsThis textbook survival guide was created for the textbook: Business Math, , edition: 9. Chapter 52: USING EQUATIONS TO SOLVE PROBLEMS includes 16 full stepbystep solutions. Since 16 problems in chapter 52: USING EQUATIONS TO SOLVE PROBLEMS have been answered, more than 19376 students have viewed full stepbystep solutions from this chapter. Business Math, was written by and is associated to the ISBN: 9780135108178. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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].

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Iterative method.
A sequence of steps intended to approach the desired solution.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).