 7.3.39: 72153445
 7.3.40: 14539489
 7.3.41: In Exercises 4144, use the inverse matrix found in Exercise15 to so...
 7.3.42: In Exercises 4144, use the inverse matrix found in Exercise15 to so...
 7.3.43: In Exercises 4144, use the inverse matrix found in Exercise15 to so...
 7.3.44: In Exercises 4144, use the inverse matrix found in Exercise15 to so...
 7.3.45: In Exercises 45 and 46, use the inverse matrix found inExercise 19 ...
 7.3.46: In Exercises 45 and 46, use the inverse matrix found inExercise 19 ...
 7.3.47: In Exercises 47 and 48, use the inverse matrix found inExercise 34 ...
 7.3.48: In Exercises 47 and 48, use the inverse matrix found inExercise 34 ...
 7.3.49: Exercises 49 and 50, use a graphing utility to solve thesystem of l...
 7.3.50: Exercises 49 and 50, use a graphing utility to solve thesystem of l...
 7.3.51: In Exercises 5158, use an inverse matrix to solve (if possible)the ...
 7.3.52: In Exercises 5158, use an inverse matrix to solve (if possible)the ...
 7.3.53: In Exercises 5158, use an inverse matrix to solve (if possible)the ...
 7.3.54: In Exercises 5158, use an inverse matrix to solve (if possible)the ...
 7.3.55: In Exercises 5158, use an inverse matrix to solve (if possible)the ...
 7.3.56: In Exercises 5158, use an inverse matrix to solve (if possible)the ...
 7.3.57: In Exercises 5158, use an inverse matrix to solve (if possible)the ...
 7.3.58: In Exercises 5158, use an inverse matrix to solve (if possible)the ...
 7.3.59: Exercises 5962, use the matrix capabilities of a graphingutility to...
 7.3.60: Exercises 5962, use the matrix capabilities of a graphingutility to...
 7.3.61: Exercises 5962, use the matrix capabilities of a graphingutility to...
 7.3.62: Exercises 5962, use the matrix capabilities of a graphingutility to...
 7.3.63: INVESTMENT PORTFOLIO In Exercises 6366, consider aperson who invest...
 7.3.64: INVESTMENT PORTFOLIO In Exercises 6366, consider aperson who invest...
 7.3.65: INVESTMENT PORTFOLIO In Exercises 6366, consider aperson who invest...
 7.3.66: INVESTMENT PORTFOLIO In Exercises 6366, consider aperson who invest...
 7.3.67: In Exercises 6770, a small home businesscreates muffins, bones, and...
 7.3.68: In Exercises 6770, a small home businesscreates muffins, bones, and...
 7.3.69: In Exercises 6770, a small home businesscreates muffins, bones, and...
 7.3.70: In Exercises 6770, a small home businesscreates muffins, bones, and...
 7.3.71: COFFEE A coffee manufacturer sells a 10poundpackage that contains ...
 7.3.72: FLOWERS A florist is creating 10 centerpieces for thetables at a we...
 7.3.73: ENROLLMENT The table shows the enrollmentprojections (in millions) ...
 7.3.74: CAPSTONE If is a matrixthen is invertible if and only if Ifverify t...
 7.3.75: TRUE OR FALSE? In Exercises 75 and 76, determinewhether the stateme...
 7.3.76: TRUE OR FALSE? In Exercises 75 and 76, determinewhether the stateme...
 7.3.77: WRITING Explain how to determine whether theinverse of a matrix exi...
 7.3.78: WRITING Explain in your own words how to write asystem of three lin...
 7.3.79: Consider matrices of the form(a) Write a matrix and a matrix in the...
Solutions for Chapter 7.3: The Inverse of a Square Matrix
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 7.3: The Inverse of a Square Matrix
Get Full SolutionsSince 41 problems in chapter 7.3: The Inverse of a Square Matrix have been answered, more than 30585 students have viewed full stepbystep solutions from this chapter. Chapter 7.3: The Inverse of a Square Matrix includes 41 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 8. College Algebra was written by and is associated to the ISBN: 9781439048696.

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

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.

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.

Elimination.
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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(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 .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

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.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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).

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.