- 6.4.1: Use Definition 6.15 to compute the determinants of the following ma...
- 6.4.2: Use Definition 6.15 to compute the determinants of the following ma...
- 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 of that make the following matrix singular. A = 1 1...
- 6.4.6: Find all values of that make the following matrix singular. A = 1 2...
- 6.4.7: Find all values of so that the following linear system has no solut...
- 6.4.8: Find all values of so that the following linear system has an infin...
- 6.4.9: Use mathematical induction to show that when n > 1, the evaluation ...
- 6.4.10: Let A be a 3 3 matrix. Show that if A is the matrix obtained from A...
- 6.4.11: Prove that AB is nonsingular if and only if both A and B are nonsin...
- 6.4.12: The solution by Cramers rule to the linear system a11x1 + a12x2 + a...
- 6.4.13: a. Generalize Cramers rule to an n n linear system. b. Use the resu...
Solutions for Chapter 6.4: The Determinant of a Matrix
Full solutions for Numerical Analysis | 9th Edition
Remove row i and column j; multiply the determinant by (-I)i + j •
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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 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!).
A symmetric matrix with eigenvalues of both signs (+ and - ).
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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).
Every v in V is orthogonal to every w in W.
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 •
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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+ (Moore-Penrose 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).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.