- 5-3.1: Hershey Foods stock earned $151,000,000. If these earnings represen...
- 5-3.2: A scale drawing of an office building is not labeled, but indicates...
- 5-3.3: A recipe uses 3 cups of flour to cups of milk. If you have 2 cups o...
- 5-3.4: For 32 hours of work, you are paid $241.60. How much would you rece...
- 5-3.5: The annual real estate tax on a duplex house is $2,321 and the owne...
- 5-3.6: A wholesale price list shows that 18 dozen headlights cost $702. If...
- 5-3.7: Two part-time employees share one full-time job. Charris works Mond...
- 5-3.8: A car that leases for $5,400 annually is leased for 8 months of the...
- 5-3.9: If 1.0000 U.S. dollar is equivalent to 0.1273 Chinese yuan, convert...
- 5-3.10: Asuntas Candle Store ordered 750 candles at a total wholesale cost ...
- 5-3.11: Sears purchased 10,000 pairs of mens slacks for $18.46 a pair and m...
- 5-3.12: K-Mart had 896 swimsuits that were marked to sell at $49.99 per uni...
Solutions for Chapter 5-3: FORMULAS
Full solutions for Business Math, | 9th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
A = CTC = (L.J]))(L.J]))T for positive definite A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).