 6.4.1: Use Definition 6.15 to compute the determinants ofthe following mat...
 6.4.2: Use Definition 6.15 to compute the determinants ofthe following mat...
 6.4.3: Repeat Exercise 1 using the method of Example 2
 6.4.4: Repeat Exercise 2 using the method of Example 2.
 6.4.5: Find all values ofa that make the following matrix singular. A = 1 ...
 6.4.6: Find all values ofa that make the following matrix singular. A = 1 ...
 6.4.7: Find all values ofa so that the following linear system has no solu...
 6.4.8: Find all values ofa so that the following linear system has an infi...
 6.4.9: The rotation matrix R = cosy sinf sin0 cost 10. applied to the vect...
 6.4.10: The rotation matrix for a 3 dimensional counterclockwise rotation t...
 6.4.11: The chemical formula x\\C(i(OH)2] +X2[HNO3] > xj\CA(N03)2] T^4!H2C]...
 6.4.12: Use mathematical induction to show that when n > 1, the evaluation ...
 6.4.13: Let A be a 3 x 3 matrix. Show that if A is the matrix obtained from...
 6.4.14: Prove that A 6 is nonsingular if and only if both A and B are nonsi...
 6.4.15: The solution by Cramer's rule to the linear system a\\X\ + 012*2 + ...
 6.4.16: Generalize Cramer's rule to an n x n linear system. Use the result ...
Solutions for Chapter 6.4: The Determinant of a Matrix
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 6.4: The Determinant of a Matrix
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.