- 5.2.1: Calculate the following determinants using cofactors. a. det 1 3 5 ...
- 5.2.2: Let A = 1 2 1 2 3 0 1 4 2 . a. If Ax = 1 2 1 , use Cramers Rule to ...
- 5.2.3: Using cofactors, find the determinant and the inverse of the matrix...
- 5.2.4: Check that Proposition 2.4 gives the customary answer for the inver...
- 5.2.5: For each of the following matrices A, calculate det(A tI ). a. _ 1 ...
- 5.2.6: Show that if the entries of a matrix A are integers, then det A is ...
- 5.2.7: a. Suppose A is an n n matrix with integer entries and det A = 1. S...
- 5.2.8: We call the vector x Rn integral if every component xi is an intege...
- 5.2.9: Prove that the exchange of any pair of rows of a matrix can be acco...
- 5.2.10: Suppose A is an orthogonal n n matrix. Show that the cofactor matri...
- 5.2.11: Generalizing the result of Proposition 2.4, show thatACT = (det A)I...
- 5.2.12: a. If C is the cofactor matrix of A, give a formula for det C in te...
- 5.2.13: a. Show that if (x1, y1) and (x2, y2) are distinct points in R2, th...
- 5.2.14: As we saw in Exercises 1.6.7 and 1.6.11, through any three noncolli...
- 5.2.15: (from the 1994 Putnam Exam) Let A and B be 2 2 matrices with intege...
- 5.2.16: In this problem, let D(x, y) denote the determinant of the 2 2 matr...
- 5.2.17: Using Exercise 16, prove that the perpendicular bisectors of the si...
Solutions for Chapter 5.2: Cofactors and Cramers Rule
Full solutions for Linear Algebra: A Geometric Approach | 2nd Edition
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).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.