 53.1: Hershey Foods stock earned $151,000,000. If these earnings represen...
 53.2: A scale drawing of an office building is not labeled, but indicates...
 53.3: A recipe uses 3 cups of flour to cups of milk. If you have 2 cups o...
 53.4: For 32 hours of work, you are paid $241.60. How much would you rece...
 53.5: The annual real estate tax on a duplex house is $2,321 and the owne...
 53.6: A wholesale price list shows that 18 dozen headlights cost $702. If...
 53.7: Two parttime employees share one fulltime job. Charris works Mond...
 53.8: A car that leases for $5,400 annually is leased for 8 months of the...
 53.9: If 1.0000 U.S. dollar is equivalent to 0.1273 Chinese yuan, convert...
 53.10: Asuntas Candle Store ordered 750 candles at a total wholesale cost ...
 53.11: Sears purchased 10,000 pairs of mens slacks for $18.46 a pair and m...
 53.12: KMart had 896 swimsuits that were marked to sell at $49.99 per uni...
Solutions for Chapter 53: FORMULAS
Full solutions for Business Math,  9th Edition
ISBN: 9780135108178
Solutions for Chapter 53: FORMULAS
Get Full SolutionsSince 12 problems in chapter 53: FORMULAS have been answered, more than 19264 students have viewed full stepbystep solutions from this chapter. Chapter 53: FORMULAS includes 12 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Business Math, was written by and is associated to the ISBN: 9780135108178. This textbook survival guide was created for the textbook: Business Math, , edition: 9.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complex conjugate
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.

Fundamental Theorem.
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 AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Norm
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 = MI AM has the same eigenvalues as A.

Spanning set.
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.

Stiffness matrix
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 = UI.
Orthonormal columns (complex analog of Q).