 4.4.4.4.1: Solve each system of linear equations using matrices. See Example 1...
 4.4.4.4.2: Solve each system of linear equations using matrices. See Example 1...
 4.4.4.4.3: Solve each system of linear equations using matrices. See Example 1...
 4.4.4.4.4: Solve each system of linear equations using matrices. See Example 1...
 4.4.4.4.5: Solve each system of linear equations using matrices. See Example 2...
 4.4.4.4.6: Solve each system of linear equations using matrices. See Example 2...
 4.4.4.4.7: Solve each system of linear equations using matrices. See Example 2...
 4.4.4.4.8: Solve each system of linear equations using matrices. See Example 2...
 4.4.4.4.9: Solve each system of linear equations using matrices. See Example 3...
 4.4.4.4.10: Solve each system of linear equations using matrices. See Example 3...
 4.4.4.4.11: Solve each system of linear equations using matrices. See Example 3...
 4.4.4.4.12: Solve each system of linear equations using matrices. See Example 3...
 4.4.4.4.13: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.14: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.15: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.16: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.17: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.18: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.19: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.20: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.21: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.22: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.23: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.24: Solve each system of linear equations using matrices. See Examples ...
 4.4.4.4.25: Determine whether each graph is the graph of a function. See Sectio...
 4.4.4.4.26: Determine whether each graph is the graph of a function. See Sectio...
 4.4.4.4.27: Determine whether each graph is the graph of a function. See Sectio...
 4.4.4.4.28: Determine whether each graph is the graph of a function. See Sectio...
 4.4.4.4.29: Evaluate. See Section 1.3.1 121 52  162132
 4.4.4.4.30: Evaluate. See Section 1.3.1221 82  1 42112
 4.4.4.4.31: Evaluate. See Section 1.3.1421102  122122
 4.4.4.4.32: Evaluate. See Section 1.3.1 72132  1 221 62
 4.4.4.4.33: Evaluate. See Section 1.3.132132  112192
 4.4.4.4.34: Evaluate. See Section 1.3.152162  11021102
 4.4.4.4.35: Solve. See the Concept Check in the sectionFor the system x + z = 7...
 4.4.4.4.36: Solve. See the Concept Check in the section.For the system ex  6 =...
 4.4.4.4.37: The amount of electricity y generated by geothermal sources (in bil...
 4.4.4.4.38: The most popular amusement park in the world (according to attendan...
 4.4.4.4.39: For the system e2x  3y = 8x + 5y = 3, explain what is wrong withw...
 4.4.4.4.40: For the system e 5x + 2y = 0y = 2, explain what is wrong withwriti...
Solutions for Chapter 4.4: Solving Systems of Equations by Matrices
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 4.4: Solving Systems of Equations by Matrices
Get Full SolutionsThis textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. Since 40 problems in chapter 4.4: Solving Systems of Equations by Matrices have been answered, more than 61828 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.4: Solving Systems of Equations by Matrices includes 40 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Column space C (A) =
space of all combinations of the columns of 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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

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

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 .

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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