 3.5.1: 2 5 4 2 3 2 = #  # =  = The value of this secondorder __________...
 3.5.2: Using Cramers rule to solve e x + y = 8 x  y = 2, we obtain x = 2...
 3.5.3: 3 2 1 4 3 1 5 1 1 3 = 3 2 2  4 2 2 + 5 2 2
 3.5.4: Using Cramers rule to solve 3x + y + 4z = 8 2x + 3y  2z = 11 x  ...
 3.5.5: 2 7 14 2 4
 3.5.6: 2 1 3 8 2
 3.5.7: 2 5 1 2 7 2
 3.5.8: 2 1 5 1 6 6 5 2
 3.5.9: 1 2 1 2 1 8  3
 3.5.10: 2 3 1 3 1 2 3 4 2
 3.5.11: e x + y = 7 x  y = 3
 3.5.12: e 2x + y = 3 x  y = 3
 3.5.13: e 12x + 3y = 15 2x  3y = 13
 3.5.14: e x  2y = 5 5x  y = 2
 3.5.15: e 4x  5y = 17 2x + 3y = 3
 3.5.16: e 3x + 2y = 2 2x + 2y = 3
 3.5.17: e x  3y = 4 3x  4y = 12
 3.5.18: e 2x  9y = 5 3x  3y = 11
 3.5.19: e 3x  4y = 4 2x + 2y = 12
 3.5.20: e 3x = 7y + 1 2x = 3y  1
 3.5.21: e 2x = 3y + 2 5x = 51  4y
 3.5.22: e y = 4x + 2 2x = 3y + 8
 3.5.23: e 3x = 2  3y 2y = 3  2x
 3.5.24: e x + 2y  3 = 0 12 = 8y + 4x
 3.5.25: e 4y = 16  3x 6x = 32  8y
 3.5.26: e 2x = 7 + 3y 4x  6y = 3
 3.5.27: 3 3 0 0 2 1 5 2 5 1 3
 3.5.28: 4 0 0 3 1 4 2 3 5
 3.5.29: 3 3 1 0 3 4 0 1 3 5
 3.5.30: 3 2 4 2 1 0 5 3 0 4
 3.5.31: 3 1 1 1 2 2 2 3 4 5
 3.5.32: 1 2 3 2 2 3 3 2 1
 3.5.33: x + y + z = 0 2x  y + z = 1 x + 3y  z = 8
 3.5.34: x  y + 2z = 3 2x + 3y + z = 9 x  y + 3z = 11
 3.5.35: 4x  5y  6z = 1 x  2y  5z = 12 2x  y = 7
 3.5.36: x  3y + z = 2 x + 2y = 8 2x  y = 1
 3.5.37: x + y + z = 4 x  2y + z = 7 x + 3y + 2z = 4
 3.5.38: 2x + 2y + 3z = 10 4x  y + z = 5 5x  2y + 6z = 1
 3.5.39: x + 2z = 4 2y  z = 5 2x + 3y = 13
 3.5.40: 3x + 2z = 4 5x  y = 4 4y + 3z = 22
 3.5.41: 2 3 1 2 3 2 2 7 0 1 5 2 2 3 0 0 7 2 2 9 6 3 5 2
 3.5.42: 2 5 0 4 3 2 2 1 0 0 1 2 2 7 5 4 6 2 2 4 1 3 5 2
 3.5.43: D = 2 2 4 3 5 2 , Dx = 2 8 4 10 5 2
 3.5.44: D = 2 2 3 5 6 2 , Dx = 2 8 3 11 6 2
 3.5.45: 2 2 x 4 6 2 = 32
 3.5.46: x + 3 6 x  2 4 2 = 28
 3.5.47: 3 1 x 2 3 1 1 0 2 2 3 = 8
 3.5.48: 3 2 x 1 3 1 0 2 1 4 3 = 39
 3.5.49: Use determinants to find the area of the triangle whose vertices ar...
 3.5.50: Use determinants to find the area of the triangle whose vertices ar...
 3.5.51: Are the points (3, 1), (0, 3), and (12, 5) collinear?
 3.5.52: Are the points (4, 6), (1, 0), and (11, 12) collinear?
 3.5.53: Use the determinant to write an equation for the line passing throu...
 3.5.54: Use the determinant to write an equation for the line passing throu...
 3.5.55: Explain how to evaluate a secondorder determinant.
 3.5.56: Describe the determinants D, Dx , and Dy in terms of the coefficien...
 3.5.57: Explain how to evaluate a thirdorder determinant.
 3.5.58: When expanding a determinant by minors, when is it necessary to sup...
 3.5.59: Without going into too much detail, describe how to solve a linear ...
 3.5.60: In applying Cramers rule, what does it mean if D = 0?
 3.5.61: The process of solving a linear system in three variables using Cra...
 3.5.62: If you could use only one method to solve linear systems in three v...
 3.5.63: Use the feature of your graphing utility that evaluates the determi...
 3.5.64: What is the fastest method for solving a linear system with your gr...
 3.5.65: Im solving a linear system using a determinant that contains two ro...
 3.5.66: I can speed up the tedious computations required by Cramers rule by...
 3.5.67: When using Cramers rule to solve a linear system, the number of det...
 3.5.68: Using Cramers rule to solve a linear system, I found the value of D...
 3.5.69: Only one 2 * 2 determinant is needed to evaluate 3 2 3 2 0 1 3 0 4 1
 3.5.70: If D = 0, then every variable has a value of 0.
 3.5.71: Because there are different determinants in the numerators of x and...
 3.5.72: Using Cramers rule, we use D Dy to get the value of y.
 3.5.73: What happens to the value of a secondorder determinant if the two ...
 3.5.74: Consider the system e a1x + b1y = c1 a2x + b2y = c2. Use Cramers ru...
 3.5.75: Show that the equation of a line through (x1 , y1) and (x2 , y2) is...
 3.5.76: Solve: 6x  4 = 2 + 6(x  1). (Section 1.4, Example 6)
 3.5.77: Solve for y: 2x + 3y = 7. (Section 1.5, Example 6)
 3.5.78: Solve: 4x + 1 3 = x  3 6 + x + 5 6 . (Section 1.4, Example 4)
 3.5.79: Solve: x + 3 4 = x  2 3 + 1 4 .
 3.5.80: Solve: 2x  4 = x + 5.
 3.5.81: Use interval notation to describe values of x for which 2(x + 4) is...
Solutions for Chapter 3.5: Determinants and Cramers Rule
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 3.5: Determinants and Cramers Rule
Get Full SolutionsThis textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Since 81 problems in chapter 3.5: Determinants and Cramers Rule have been answered, more than 7097 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Chapter 3.5: Determinants and Cramers Rule includes 81 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Outer product uv T
= column times row = rank one matrix.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Textbook Survival Guides
I don't want to reset my password
Need help? Contact support
Having trouble accessing your account? Let us help you, contact support at +1(510) 9441054 or support@studysoup.com
Forgot password? Reset it here