- 5-2.1: Find the value of the variable:
- 5-2.2: Find the value of the variable:
- 5-2.3: Find the value of the variable:
- 5-2.4: Find the value of the variable:
- 5-2.5: Find the value of the variable:
- 5-2.6: Find the value of the variable:
- 5-2.7: Find the value of the variable:
- 5-2.8: Find the value of the variable:
- 5-2.9: Find the value of the variable:
- 5-2.10: Find the value of the variable:
- 5-2.11: The difference in hours between full-timers and the parttimers who ...
- 5-2.12: Manny plans to save of his salary each week. If his weekly salary i...
- 5-2.13: Last week at the Sunshine Valley Rock Festival, Joel sold 3 times a...
- 5-2.14: Elaine sold 3 times as many magazine subscriptions as Ron did. Ron ...
- 5-2.15: Will ordered 2 times as many boxes of ballpoint pens as boxes of fe...
- 5-2.16: A real estate salesperson bought promotional calendars and date boo...
Solutions for Chapter 5-2: USING EQUATIONS TO SOLVE PROBLEMS
Full solutions for Business Math, | 9th Edition
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.
Remove row i and column j; multiply the determinant by (-I)i + j •
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 n-l - 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 Fn-1c 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.
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 = Q-l. 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 = M-I 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.
Skew-symmetric 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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).