 7.4.45: 131424203
 7.4.46: 211140342
 7.4.47: 200430615
 7.4.48: 3710112002
 7.4.49: 2213675763002617
 7.4.50: 3210601356214021
 7.4.51: 54003621043261242
 7.4.52: 1503460232012105
 7.4.53: 3216320000410251341152000
 7.4.54: 5000021000042300364022312
 7.4.55: In Exercises 5562, use the matrix capabilities of a graphingutility...
 7.4.56: In Exercises 5562, use the matrix capabilities of a graphingutility...
 7.4.57: In Exercises 5562, use the matrix capabilities of a graphingutility...
 7.4.58: In Exercises 5562, use the matrix capabilities of a graphingutility...
 7.4.59: In Exercises 5562, use the matrix capabilities of a graphingutility...
 7.4.60: In Exercises 5562, use the matrix capabilities of a graphingutility...
 7.4.61: In Exercises 5562, use the matrix capabilities of a graphingutility...
 7.4.62: In Exercises 5562, use the matrix capabilities of a graphingutility...
 7.4.63: In Exercises 6370, find (a) (b) (c) and
 7.4.64: In Exercises 6370, find (a) (b) (c) and
 7.4.65: In Exercises 6370, find (a) (b) (c) and
 7.4.66: In Exercises 6370, find (a) (b) (c) and
 7.4.67: In Exercises 6370, find (a) (b) (c) and
 7.4.68: In Exercises 6370, find (a) (b) (c) and
 7.4.69: In Exercises 6370, find (a) (b) (c) and
 7.4.70: In Exercises 6370, find (a) (b) (c) and
 7.4.71: In Exercises 7176, evaluate the determinant(s) to verify theequation.
 7.4.72: In Exercises 7176, evaluate the determinant(s) to verify theequation.
 7.4.73: In Exercises 7176, evaluate the determinant(s) to verify theequation.
 7.4.74: In Exercises 7176, evaluate the determinant(s) to verify theequation.
 7.4.75: In Exercises 7176, evaluate the determinant(s) to verify theequation.
 7.4.76: In Exercises 7176, evaluate the determinant(s) to verify theequation.
 7.4.77: In Exercises 7784, solve for x x12x 2
 7.4.78: In Exercises 7784, solve for x x14x 20
 7.4.79: In Exercises 7784, solve for x 21x 2 1
 7.4.80: In Exercises 7784, solve for x x 112x 4
 7.4.81: In Exercises 7784, solve for x x 1 032x 2 0
 7.4.82: In Exercises 7784, solve for x x 231x x 1 0
 7.4.83: In Exercises 7784, solve for x x 312x 2 0
 7.4.84: x 472 x 5 0
 7.4.85: In Exercises 8590, evaluate the determinant in which theentries are...
 7.4.86: In Exercises 8590, evaluate the determinant in which theentries are...
 7.4.87: In Exercises 8590, evaluate the determinant in which theentries are...
 7.4.88: In Exercises 8590, evaluate the determinant in which theentries are...
 7.4.89: In Exercises 8590, evaluate the determinant in which theentries are...
 7.4.90: In Exercises 8590, evaluate the determinant in which theentries are...
 7.4.91: TRUE OR FALSE? In Exercises 91 and 92, determinewhether the stateme...
 7.4.92: TRUE OR FALSE? In Exercises 91 and 92, determinewhether the stateme...
 7.4.93: Find square matrices and to demonstrate that A B A B.
 7.4.94: Consider square matrices in which the entries areconsecutive intege...
 7.4.95: WRITING Write a brief paragraph explaining thedifference between a ...
 7.4.96: THINK ABOUT IT If is a matrix of order suchthat is it possible to f...
 7.4.97: In Exercises 9799,a property of determinants is given ( and are squ...
 7.4.98: In Exercises 9799,a property of determinants is given ( and are squ...
 7.4.99: In Exercises 9799,a property of determinants is given ( and are squ...
 7.4.100: CAPSTONE If is an matrix, explain how tofind the following.(a) The ...
 7.4.101: In Exercises 101104, evaluate the determinant 100050002
 7.4.102: In Exercises 101104, evaluate the determinant 2000020000100003
 7.4.103: In Exercises 101104, evaluate the determinant 100230543
 7.4.104: In Exercises 101104, evaluate the determinant 14501100 5
 7.4.105: CONJECTURE A triangular matrix is a squarematrix with all zero entr...
 7.4.106: Use the matrix capabilities of a graphing utility to find thedeterm...
Solutions for Chapter 7.4: The Determinant of a Square Matrix
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 7.4: The Determinant of a Square Matrix
Get Full SolutionsSince 62 problems in chapter 7.4: The Determinant of a Square Matrix have been answered, more than 30924 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: 8. Chapter 7.4: The Determinant of a Square Matrix includes 62 full stepbystep solutions. 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.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

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].

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.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.