- 20-3.1: Find the taxable income. Use $3,650 for each exemption. 4 $49,071 $...
- 20-3.2: Find the taxable income. Use $3,650 for each exemption. 1 $138,503 ...
- 20-3.3: Find the taxable income. Use $3,650 for each exemption. 5 $167,413 ...
- 20-3.4: Find the taxable income. Use $3,650 for each exemption. 2 $75,013 $...
- 20-3.5: Use Table 20-1 to find the federal income tax. $40,317 Single
- 20-3.6: Use Table 20-1 to find the federal income tax. $32,417 Married, fil...
- 20-3.7: Use Table 20-1 to find the federal income tax. $30,307 Married, fil...
- 20-3.8: Use Table 20-1 to find the federal income tax. $29,553 Head of hous...
- 20-3.9: Use Table 20-2 to find the federal income tax. $172,518 Single
- 20-3.10: Use Table 20-2 to find the federal income tax. $198,846 Married, fi...
- 20-3.11: Find the taxable income for a family of six (husband, wife, four ch...
- 20-3.12: Find the taxable income for a single person whose adjusted gross in...
- 20-3.13: Canty ONeal has an adjusted gross income of $68,917 and itemized de...
- 20-3.14: Noel Womack is single and calculates his taxable income to be $30,1...
- 20-3.15: Tommy and Michelle Fernandez have a combined taxable income of $23,...
- 20-3.16: Vladimir Bozin is a head of household and has a taxable income of $...
- 20-3.17: Donna Shroyer is single and has a taxable income of $29,897. If her...
- 20-3.18: Paul Smith is married and filing his tax jointly with his wife, Ann...
- 20-3.19: Dr. Steven Katz is single and has a taxable income of $160,842. Use...
- 20-3.20: Jack Falcinelli is filing his tax as a head of household. His taxab...
Solutions for Chapter 20-3: INCOME TAXES
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.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Remove row i and column j; multiply the determinant by (-I)i + j •
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
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.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.