 12.1: Mentally estimate the sum by rounding to the first digit, then add ...
 12.2: Mentally estimate the sum by rounding to the first digit, then add ...
 12.3: Mentally estimate the sum by rounding to the first digit, then add ...
 12.4: Mentally estimate the sum by rounding to the first digit, then add ...
 12.5: Add the integers. 42 + 36
 12.6: Add the integers. 283 + 375
 12.7: Add the integers. 4,216 + (3,972)
 12.8: Subtract and check the difference5 5  3 6
 12.9: Subtract and check the difference3 0 8  2 7 5
 12.10: Subtract and check the difference5,409  2,176
 12.11: Subtract the integers. 18  (3)
 12.12: Subtract the integers. 12  (5)
 12.13: Subtract the integers. 5  (17)
 12.14: Subtract the integers. 37 41
 12.15: Multiply and check the product 730 * 60
 12.16: Multiply and check the product 904 * 24
 12.17: Multiply and check the product 1,005 by 89
 12.18: Multiply the integers (3)(46)
 12.19: Multiply the integers (32)(15)
 12.20: Multiply the integers (64)(83)
 12.21: Multiply the integers ( $82,916)(7)
 12.22: Divide and check the quotient 96 , 6
 12.23: Divide and check the quotient 13,838 , 34
 12.24: Divide and check the quotient 17_4,420
 12.25: Divide the integers 72 , (9)
 12.26: Divide the integers (56) , (8)
 12.27: Divide the integers 672 , 16
 12.28: Divide the integers  $13,623 , 57
 12.29: Perform the operations by applying the standard order of operations...
 12.30: Perform the operations by applying the standard order of operations...
 12.31: Perform the operations by applying the standard order of operations...
 12.32: Perform the operations by applying the standard order of operations...
 12.33: The menswear department of the Gap has a sales goal of $1,384,000 f...
 12.34: Atkinsons Candy Company manufactures seven types of hard candy for ...
 12.35: University Trailer Sales Company sold 352 utility trailers during a...
 12.36: An acre of ground is a square piece of land that is 210 feet on eac...
 12.37: If you are paying three employees $9 per hour and the fence install...
 12.38: The 7th Inning buys baseball cards from eight vendors. In the month...
 12.39: If you have 348 packages of holiday candy to rebox for shipment to ...
 12.40: Bio Fach, Germanys biggest ecologically sound consumer goods trade ...
 12.41: The communication revolution has given us prepaid phone calling car...
 12.42: Strategic Telecomm Systems, Inc. (STS), in Knoxville, Tennessee, ma...
 12.43: In Exercise 42, if STS resells the phone time at an average of 6 ce...
 12.44: American Communications Network (ACN) of Troy, Michigan, also marke...
 12.45: Last year Wilmington Motors lost $39,583. This year the company los...
 12.46: Brentwood Fashions posted a net loss of $32,871 last year and a net...
 12.47: Lisle Building Supplies sold 291 rolls of damaged insulation at a $...
 12.48: Kent Realty Company had an annual loss of $63,408. What was the ave...
Solutions for Chapter 12: OPERATIONS WITH WHOLE NUMBERS AND INTEGERS
Full solutions for Business Math,  9th Edition
ISBN: 9780135108178
Solutions for Chapter 12: OPERATIONS WITH WHOLE NUMBERS AND INTEGERS
Get Full SolutionsSince 48 problems in chapter 12: OPERATIONS WITH WHOLE NUMBERS AND INTEGERS have been answered, more than 18442 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Business Math, , edition: 9. Business Math, was written by and is associated to the ISBN: 9780135108178. Chapter 12: OPERATIONS WITH WHOLE NUMBERS AND INTEGERS includes 48 full stepbystep solutions.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

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.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·