 7.7.7.7.1: Fill in the blanks. Both det(A) and represent the _______ of the ma...
 7.7.7.7.2: Fill in the blanks. The _______ of the entry is the determinant of ...
 7.7.7.7.3: Fill in the blanks. The _______ of the entry is given by
 7.7.7.7.4: Fill in the blanks. One way of finding the determinant of a matrix ...
 7.7.7.7.5: Fill in the blanks. A square matrix with all zero entries either ab...
 7.7.7.7.6: Fill in the blanks. A matrix that is both upper and lower triangula...
 7.7.7.7.7: In Exercises 112, find the determinant of the matrix.
 7.7.7.7.8: In Exercises 112, find the determinant of the matrix.
 7.7.7.7.9: In Exercises 112, find the determinant of the matrix.
 7.7.7.7.10: In Exercises 112, find the determinant of the matrix.
 7.7.7.7.11: In Exercises 112, find the determinant of the matrix.
 7.7.7.7.12: In Exercises 112, find the determinant of the matrix.
 7.7.7.7.13: In Exercises 13 and 14, use the matrix capabilities of a graphing u...
 7.7.7.7.14: In Exercises 13 and 14, use the matrix capabilities of a graphing u...
 7.7.7.7.15: In Exercises 1518, find all (a) minors and (b) cofactors of the matrix
 7.7.7.7.16: In Exercises 1518, find all (a) minors and (b) cofactors of the matrix
 7.7.7.7.17: In Exercises 1518, find all (a) minors and (b) cofactors of the matrix
 7.7.7.7.18: In Exercises 1518, find all (a) minors and (b) cofactors of the matrix
 7.7.7.7.19: In Exercises 1922, find the determinant of the matrix by the method...
 7.7.7.7.20: In Exercises 1922, find the determinant of the matrix by the method...
 7.7.7.7.21: In Exercises 1922, find the determinant of the matrix by the method...
 7.7.7.7.22: In Exercises 1922, find the determinant of the matrix by the method...
 7.7.7.7.23: In Exercises 23 28, find the determinant of the matrix. Expand by c...
 7.7.7.7.24: In Exercises 23 28, find the determinant of the matrix. Expand by c...
 7.7.7.7.25: In Exercises 23 28, find the determinant of the matrix. Expand by c...
 7.7.7.7.26: In Exercises 23 28, find the determinant of the matrix. Expand by c...
 7.7.7.7.27: In Exercises 23 28, find the determinant of the matrix. Expand by c...
 7.7.7.7.28: In Exercises 23 28, find the determinant of the matrix. Expand by c...
 7.7.7.7.29: In Exercises 2932, evaluate the determinant. Do not use a graphing ...
 7.7.7.7.30: In Exercises 2932, evaluate the determinant. Do not use a graphing ...
 7.7.7.7.31: In Exercises 2932, evaluate the determinant. Do not use a graphing ...
 7.7.7.7.32: In Exercises 2932, evaluate the determinant. Do not use a graphing ...
 7.7.7.7.33: In Exercises 3336, use the matrix capabilities of a graphing utilit...
 7.7.7.7.34: In Exercises 3336, use the matrix capabilities of a graphing utilit...
 7.7.7.7.35: In Exercises 3336, use the matrix capabilities of a graphing utilit...
 7.7.7.7.36: In Exercises 3336, use the matrix capabilities of a graphing utilit...
 7.7.7.7.37: In Exercises 37 40, find (a) (b) (c) and (d)
 7.7.7.7.38: In Exercises 37 40, find (a) (b) (c) and (d)
 7.7.7.7.39: In Exercises 37 40, find (a) (b) (c) and (d)
 7.7.7.7.40: In Exercises 37 40, find (a) (b) (c) and (d)
 7.7.7.7.41: In Exercises 41 and 42, use the matrix capabilities of a graphing u...
 7.7.7.7.42: In Exercises 41 and 42, use the matrix capabilities of a graphing u...
 7.7.7.7.43: In Exercises 43 48, evaluate the determinants to verify the equation.
 7.7.7.7.44: In Exercises 43 48, evaluate the determinants to verify the equation.
 7.7.7.7.45: In Exercises 43 48, evaluate the determinants to verify the equation.
 7.7.7.7.46: In Exercises 43 48, evaluate the determinants to verify the equation.
 7.7.7.7.47: In Exercises 43 48, evaluate the determinants to verify the equation.
 7.7.7.7.48: In Exercises 43 48, evaluate the determinants to verify the equation.
 7.7.7.7.49: In Exercises 4960, solve for x.
 7.7.7.7.50: In Exercises 4960, solve for x.
 7.7.7.7.51: In Exercises 4960, solve for x.
 7.7.7.7.52: In Exercises 4960, solve for x.
 7.7.7.7.53: In Exercises 4960, solve for x.
 7.7.7.7.54: In Exercises 4960, solve for x.
 7.7.7.7.55: In Exercises 4960, solve for x.
 7.7.7.7.56: In Exercises 4960, solve for x.
 7.7.7.7.57: In Exercises 4960, solve for x.
 7.7.7.7.58: In Exercises 4960, solve for x.
 7.7.7.7.59: In Exercises 4960, solve for x.
 7.7.7.7.60: In Exercises 4960, solve for x.
 7.7.7.7.61: In Exercises 6166, evaluate the determinant, in which the entries a...
 7.7.7.7.62: In Exercises 6166, evaluate the determinant, in which the entries a...
 7.7.7.7.63: In Exercises 6166, evaluate the determinant, in which the entries a...
 7.7.7.7.64: In Exercises 6166, evaluate the determinant, in which the entries a...
 7.7.7.7.65: In Exercises 6166, evaluate the determinant, in which the entries a...
 7.7.7.7.66: In Exercises 6166, evaluate the determinant, in which the entries a...
 7.7.7.7.67: True or False? In Exercises 67 and 68, determine whether the statem...
 7.7.7.7.68: True or False? In Exercises 67 and 68, determine whether the statem...
 7.7.7.7.69: Exploration Find square matrices and to demonstrate that
 7.7.7.7.70: Conjecture Consider square matrices in which the entries are consec...
 7.7.7.7.71: In Exercises 7174, (a) find the determinant of A, (b) find (c) find...
 7.7.7.7.72: In Exercises 7174, (a) find the determinant of A, (b) find (c) find...
 7.7.7.7.73: In Exercises 7174, (a) find the determinant of A, (b) find (c) find...
 7.7.7.7.74: In Exercises 7174, (a) find the determinant of A, (b) find (c) find...
 7.7.7.7.75: In Exercises 7577, a property of determinants is given (A and B are...
 7.7.7.7.76: In Exercises 7577, a property of determinants is given (A and B are...
 7.7.7.7.77: In Exercises 7577, a property of determinants is given (A and B are...
 7.7.7.7.78: Writing Write an argument that explains why the determinant of a tr...
 7.7.7.7.79: In Exercises 79 82, factor the expression.
 7.7.7.7.80: In Exercises 79 82, factor the expression.
 7.7.7.7.81: In Exercises 79 82, factor the expression.
 7.7.7.7.82: In Exercises 79 82, factor the expression.
 7.7.7.7.83: In Exercises 83 and 84, solve the system of equations using the met...
 7.7.7.7.84: In Exercises 83 and 84, solve the system of equations using the met...
Solutions for Chapter 7.7: The Determinant of a Square Matrix
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 7.7: The Determinant of a Square Matrix
Get Full SolutionsPrecalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 7.7: The Determinant of a Square Matrix includes 84 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Since 84 problems in chapter 7.7: The Determinant of a Square Matrix have been answered, more than 36202 students have viewed full stepbystep solutions from this chapter.

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

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

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

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.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

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

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