- 2.2.1: Evaluate each of the following determinants by inspection: (a) ____...
- 2.2.2: Let A = 0 1 2 3 1 1 1 1 2 2 3 3 1 2 2 3 (a) Use the elimination met...
- 2.2.3: For each of the following, compute the determinant and state whethe...
- 2.2.4: Find all possible choices of c that would make the following matrix...
- 2.2.5: Let A be an n n matrix and a scalar. Show that det(A) = n det(A)
- 2.2.6: Let A be a nonsingular matrix. Show that det(A1) = 1 det(A)
- 2.2.7: Let A and B be 33 matrices with det(A) = 4 and det(B) = 5. Find the...
- 2.2.8: Show that if E is an elementary matrix, then ET is an elementary ma...
- 2.2.9: Let E1, E2, and E3 be 33 elementary matrices of types I, II, and II...
- 2.2.10: Let A and B be row equivalent matrices, and suppose that B can be o...
- 2.2.11: Let A be an n n matrix. Is it possible for A2+I = O in the case whe...
- 2.2.12: Consider the 3 3 Vandermonde matrix V = 1 x1 x2 1 1 x2 x2 2 1 x3 x2...
- 2.2.13: Suppose that a 33 matrix A factors into a product 1 0 0 l21 1 0 l31...
- 2.2.14: Let A and B be nn matrices. Prove that the product AB is nonsingula...
- 2.2.15: Let A and B be n n matrices. Prove that if AB = I , then BA = I . W...
- 2.2.16: A matrix A is said to be skew symmetric if AT = A. For example, A =...
- 2.2.17: Let A be a nonsingular nn matrix with a nonzero cofactor Ann, and s...
- 2.2.18: Let A be a k k matrix and let B be an (n k) (n k) matrix. Let E = I...
- 2.2.19: Let A and B be k k matrices and let M = O B A O Show that det(M) = ...
- 2.2.20: Show that evaluating the determinant of an n n matrix by cofactors ...
- 2.2.21: Show that the elimination method of computing the value of the dete...
Solutions for Chapter 2.2: Properties of Determinants
Full solutions for Linear Algebra with Applications | 8th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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).
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Invert A by row operations on [A I] to reach [I A-I].
A symmetric matrix with eigenvalues of both signs (+ and - ).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Every v in V is orthogonal to every w in W.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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 r (A)
= number of pivots = dimension of column space = dimension of row space.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.