 6.5.1: Evaluate each determinant in Exercises 110. I;
 6.5.2: Evaluate each determinant in Exercises 110. ~ :
 6.5.3: Evaluate each determinant in Exercises 110. 14 5 ~I
 6.5.4: Evaluate each determinant in Exercises 110. ~ ~
 6.5.5: Evaluate each determinant in Exercises 110. n
 6.5.6: Evaluate each determinant in Exercises 110. ~
 6.5.7: Evaluate each determinant in Exercises 110. 15 2_,1  ?
 6.5.8: Evaluate each determinant in Exercises 110. ! !I
 6.5.9: Evaluate each determinant in Exercises 110. i II
 6.5.10: Evaluate each determinant in Exercises 110. 1f t
 6.5.11: For Exercises 1122, use Cramer's Rule to solve each system. {x + y...
 6.5.12: For Exercises 1122, use Cramer's Rule to solve each system. 2r + y...
 6.5.13: For Exercises 1122, use Cramer's Rule to solve each system. 2r + 3...
 6.5.14: For Exercises 1122, use Cramer's Rule to solve each system. x  2y...
 6.5.15: For Exercises 1122, use Cramer's Rule to solve each system. {4x  ...
 6.5.16: For Exercises 1122, use Cramer's Rule to solve each system. fx + 2...
 6.5.17: For Exercises 1122, use Cramer's Rule to solve each system. X+ ? ...
 6.5.18: For Exercises 1122, use Cramer's Rule to solve each system. {2r  ...
 6.5.19: For Exercises 1122, use Cramer's Rule to solve each system. ex . ...
 6.5.20: For Exercises 1122, use Cramer's Rule to solve each system. >r  ?...
 6.5.21: For Exercises 1122, use Cramer's Rule to solve each system. 2r  3...
 6.5.22: For Exercises 1122, use Cramer's Rule to solve each system. y   ...
 6.5.23: Evaluate each determinant in Exercises 2328. 3 0 0 2 1  5 2 5  1
 6.5.24: Evaluate each determinant in Exercises 2328. 4 0 0 3  1 4 2  3 5
 6.5.25: Evaluate each determinant in Exercises 2328. 3 1 0  3 4 0  1 3  5
 6.5.26: Evaluate each determinant in Exercises 2328. 2 4 2  I 0 5 3 0 4
 6.5.27: Evaluate each determinant in Exercises 2328. 1 1 1 2 2 2  3 4  5
 6.5.28: Evaluate each determinant in Exercises 2328. 1 2 32 2  33 2 1
 6.5.29: In Exercises 2936. use Cramer's Rule to solve each system. x+ y +z...
 6.5.30: In Exercises 2936. use Cramer's Rule to solve each system. x  y +...
 6.5.31: In Exercises 2936. use Cramer's Rule to solve each system. 4x  5y...
 6.5.32: In Exercises 2936. use Cramer's Rule to solve each system. x  3y ...
 6.5.33: In Exercises 2936. use Cramer's Rule to solve each system. x + y +...
 6.5.34: In Exercises 2936. use Cramer's Rule to solve each system. {2r + 2...
 6.5.35: In Exercises 2936. use Cramer's Rule to solve each system. .t+ 2z ...
 6.5.36: In Exercises 2936. use Cramer's Rule to solve each system. 3x+2z ...
 6.5.37: Evaluate each determinam in Exercises 3740. 4 2 8  ?  2 0 4  2 ...
 6.5.38: Evaluate each determinam in Exercises 3740. 3  1 1 2  2 0 0 0 2...
 6.5.39: Evaluate each determinam in Exercises 3740.  2  3 3 5 1  4 0 0 ...
 6.5.40: Evaluate each determinam in Exercises 3740. 1  3 2 0 3  1 0  2...
 6.5.41: In Exercises 4142, evaluate each determinant. 3 11 17 01  2 3 1 5...
 6.5.42: In Exercises 4142, evaluate each determinant. 01 1l 01 4  I~ ~I ...
 6.5.43: In Exercises 4344, write the system of linear equations for which ...
 6.5.44: In Exercises 4344, write the system of linear equations for which ...
 6.5.45: In Exercises 4548, solve each equation for x. ~ ~I 32
 6.5.46: In Exercises 4548, solve each equation for x. I; ~ ~ 61 28  4
 6.5.47: In Exercises 4548, solve each equation for x. 1 X  23 1 1   80 ...
 6.5.48: In Exercises 4548, solve each equation for x. 2 X 3261 I0  394
 6.5.49: Determinant.s are used to find the arf!a of a triangle whose vertic...
 6.5.50: Determinant.s are used to find the arf!a of a triangle whose vertic...
 6.5.51: Determinants are used to show that three pohJts lie on the same lin...
 6.5.52: Determinants are used to show that three pohJts lie on the same lin...
 6.5.53: Dett!rminants are used to write an equation of a line passing throu...
 6.5.54: Dett!rminants are used to write an equation of a line passing throu...
 6.5.55: Explain how to evaluate a secondorder deteminant.
 6.5.56: Describe the determinants DK a nd D1 in tem1s of the coefficients a...
 6.5.57: Explain how to evaluate a thirdorder determinant.
 6.5.58: When expanding a determinant by minors, when is it necessary to sup...
 6.5.59: Without going into too much detail, describe how to solve a linear ...
 6.5.60: In applying Cramer's Rule, what should you do if D  0?
 6.5.61: The process of solving a linear syste.m in three variables using Cr...
 6.5.62: If you could use only one method to solve linear systems in three v...
 6.5.63: Use the feature of your graphing utility that evaluates the detcnni...
 6.5.64: In Exercises 6465, use a graphing utility to evaluate the dt!termi...
 6.5.65: In Exercises 6465, use a graphing utility to evaluate the dt!termi...
 6.5.66: What is the fastest method for solving a linear system with your gr...
 6.5.67: In Exercises 6770, determine whether each statement makes sense or...
 6.5.68: In Exercises 6770, determine whether each statement makes sense or...
 6.5.69: In Exercises 6770, determine whether each statement makes sense or...
 6.5.70: In Exercises 6770, determine whether each statement makes sense or...
 6.5.71: a. Evaluate: 0 a a a b. Evaluate: 0 a a . 0 0 (I a a a a Evaluate: ...
 6.5.72: Evaluate: 2 0 0 0 0 0 3 0 0 0 0 0 2 0 0. 0 0 0 I 0 0 0 0 0 4
 6.5.73: What happens to the value of a secondorder determinant if the two ...
 6.5.74: Consider the system { a1x + bly  c1 ar + b~} Cz . Use Cramers Rule...
 6.5.75: Show that the equation of a line through (x1 ,y1) and (x.,, y2) is ...
 6.5.76: We have seen that determinants can be used to solve linear equation...
 6.5.77: Exercises 7779 willl help you prepare for the nwter;at covered in ...
 6.5.78: Exercises 7779 willl help you prepare for the nwter;at covered in ...
 6.5.79: Exercises 7779 willl help you prepare for the nwter;at covered in ...
Solutions for Chapter 6.5: Determinants and Cramer's Rule
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 6.5: Determinants and Cramer's Rule
Get Full SolutionsChapter 6.5: Determinants and Cramer's Rule includes 79 full stepbystep solutions. Since 79 problems in chapter 6.5: Determinants and Cramer's Rule have been answered, more than 25025 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: College Algebra , edition: 6. College Algebra was written by and is associated to the ISBN: 9780321782281.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + 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.

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

GramSchmidt 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.

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.

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.

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 N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.