 3.1: In Exercises 118, find the determinant of the matrix
 3.2: In Exercises 118, find the determinant of the matrix
 3.3: In Exercises 118, find the determinant of the matrix
 3.4: In Exercises 118, find the determinant of the matrix
 3.5: In Exercises 118, find the determinant of the matrix
 3.6: In Exercises 118, find the determinant of the matrix
 3.7: In Exercises 118, find the determinant of the matrix
 3.8: In Exercises 118, find the determinant of the matrix
 3.9: In Exercises 118, find the determinant of the matrix
 3.10: In Exercises 118, find the determinant of the matrix
 3.11: In Exercises 118, find the determinant of the matrix
 3.12: In Exercises 118, find the determinant of the matrix
 3.13: In Exercises 118, find the determinant of the matrix
 3.14: In Exercises 118, find the determinant of the matrix
 3.15: In Exercises 118, find the determinant of the matrix
 3.16: In Exercises 118, find the determinant of the matrix
 3.17: In Exercises 118, find the determinant of the matrix
 3.18: In Exercises 118, find the determinant of the matrix
 3.19: In Exercises 1922, determine which property of determinants is illu...
 3.20: In Exercises 1922, determine which property of determinants is illu...
 3.21: In Exercises 1922, determine which property of determinants is illu...
 3.22: In Exercises 1922, determine which property of determinants is illu...
 3.23: In Exercises 23 and 24, find (a) (b) (c) and (d) Then verify that
 3.24: In Exercises 23 and 24, find (a) (b) (c) and (d) Then verify that
 3.25: In Exercises 25 and 26, find (a) (b) (c) and
 3.26: In Exercises 25 and 26, find (a) (b) (c) and
 3.27: In Exercises 27 and 28, find (a) and
 3.28: In Exercises 27 and 28, find (a) and
 3.29: In Exercises 2932, find Begin by finding and then evaluate its dete...
 3.30: In Exercises 2932, find Begin by finding and then evaluate its dete...
 3.31: In Exercises 2932, find Begin by finding and then evaluate its dete...
 3.32: In Exercises 2932, find Begin by finding and then evaluate its dete...
 3.33: In Exercises 3336, solve the system of linear equations by each of ...
 3.34: In Exercises 3336, solve the system of linear equations by each of ...
 3.35: In Exercises 3336, solve the system of linear equations by each of ...
 3.36: In Exercises 3336, solve the system of linear equations by each of ...
 3.37: In Exercises 3742, use the determinant of the coefficient matrix to...
 3.38: In Exercises 3742, use the determinant of the coefficient matrix to...
 3.39: In Exercises 3742, use the determinant of the coefficient matrix to...
 3.40: In Exercises 3742, use the determinant of the coefficient matrix to...
 3.41: In Exercises 3742, use the determinant of the coefficient matrix to...
 3.42: In Exercises 3742, use the determinant of the coefficient matrix to...
 3.43: (a) The cofactor of a given matrix is always a positive number. (b)...
 3.44: (a) If and are square matrices of order such that then both and are...
 3.45: If is a matrix such that then what is the value of
 3.46: If is a matrix such that then what is the value of
 3.47: Prove the property below.
 3.48: Illustrate the property shown in Exercise 47 for the following.
 3.49: Find the determinant of the matrix.
 3.50: Show that
 3.51: In Exercises 5154, find the eigenvalues and corresponding eigenvect...
 3.52: In Exercises 5154, find the eigenvalues and corresponding eigenvect...
 3.53: In Exercises 5154, find the eigenvalues and corresponding eigenvect...
 3.54: In Exercises 5154, find the eigenvalues and corresponding eigenvect...
 3.55: Calculus In Exercises 5558, find the Jacobians of the functions. If...
 3.56: Calculus In Exercises 5558, find the Jacobians of the functions. If...
 3.57: Calculus In Exercises 5558, find the Jacobians of the functions. If...
 3.58: Calculus In Exercises 5558, find the Jacobians of the functions. If...
 3.59: Writing Compare the various methods for calculating the determinant...
 3.60: Prove that if and and are of the same size, then there exists a mat...
 3.61: In Exercises 61 and 62, find the adjoint of the matrix.
 3.62: In Exercises 61 and 62, find the adjoint of the matrix.
 3.63: In Exercises 6366, use the determinant of the coefficient matrix to...
 3.64: In Exercises 6366, use the determinant of the coefficient matrix to...
 3.65: In Exercises 6366, use the determinant of the coefficient matrix to...
 3.66: In Exercises 6366, use the determinant of the coefficient matrix to...
 3.67: The table shows the projected populations (in millions) of the Unit...
 3.68: The table shows the projected amounts (in dollars) spent per person...
 3.69: In Exercises 69 and 70, use a determinant to find the area of the t...
 3.70: In Exercises 69 and 70, use a determinant to find the area of the t...
 3.71: In Exercises 71 and 72, use the determinant to find an equation of ...
 3.72: In Exercises 71 and 72, use the determinant to find an equation of ...
 3.73: In Exercises 73 and 74, find an equation of the plane passingthroug...
 3.74: In Exercises 73 and 74, find an equation of the plane passingthroug...
 3.75: (a) In Cramers Rule, the value of is the quotient of two determinan...
 3.76: (a) If is a square matrix, then the matrix of cofactors of is calle...
Solutions for Chapter 3: Determinants
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 3: Determinants
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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.

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.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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