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# 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

Solutions for Chapter 6.5
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##### ISBN: 9780321782281

Chapter 6.5: Determinants and Cramer's Rule includes 79 full step-by-step solutions. Since 79 problems in chapter 6.5: Determinants and Cramer's Rule have been answered, more than 72144 students have viewed full step-by-step 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.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Cofactor Cij.

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

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• 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].

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Fibonacci numbers

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].

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• lA-II = 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.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• 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 .

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• 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.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.