Let A ? R m×n , let~ai = (ai1 ai2 ··· ain) be the i th row of A,~cj = ? ???? a1 j a2 j . . . am j ?...

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Let A ? R m×n , let~ai = (ai1 ai2 ··· ain) be the i th row of A,~cj = ? ???? a1 j a2 j . . . am j ? ???? the j th column of A, let En = {~e1, ~e2, ..., ~en} be the standard basis for R n , and let Em = {~e1, ~e2, ..., ~em} be the standard basis for R m. (a) Suppose detA = 1. What does this tell you about m and n? (b) Determine A~ej and~ei TA. (c) Determine~ei TA~ej . (d) Suppose~x ? R n is perpindicular to all the rows of A Determine A~x. (e) Suppose~y ? R m is perpindicular to all the columns of A. Show that A~x ·~y = 0 for all~x ? R n . (f) What relation between the number of rows m and the number of columns n will ensure that the homogeneous linear system A~x =~0 has infinitely many solutions? (g) Suppose that for some matrix B the products AB and BA are both defined. What can you say about the number of rows and columns of B? 2. Let A ? R n×n be a square matrix. As before let~ai = (ai1 ai2 ··· ain) be the i th row of A,~cj = ? ???? a1 j a2 j . . . an j ? ???? the j th column of A, and let En = {~e1, ~e2, ..., ~en} be the standard basis for R n . Suppose A is non-singular. (a) What is the span of the columns of A? (b) What is the reduced row echelon form of A? Printed on August 28, 2012 Page 1 of 2 Mathematical Economics Assignment 1 3. he characteristic polynomial p(?) of an n × n matrix A is a monic polynomial of degree n in ?. By the Fundamental Theorem of Algebra p can be completely factored into the product of n linear terms. That is, p(?) = (?-?1)(?-?2)···(?-?n), (1) where the ?i’s are the possibly repeated and possibly complex roots of p. So, if ?1, ?2, ...,?k are the distinct roots of p then, p(?) = (?-?1) m1 (?-?2) m2 ···(?-?k) mk . (2) The roots of p are the eigenvalues of A. The exponent mj of the linear factor (?-?j) is called the algebraic multiplicity of ?j . The dimension, (size of a maximal linear independent subset), of the eigenspace E?j is the geometric multiplicity of ?j . Use equations (1) and or (2) to argue that the determinant of A is the product of the all the eigenvalues raised to their algebraic multiplicity. Hint: What is p(0)? 4. Let A = ? ? ? ? ? ? XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX ? ? ? ? ? ? . Determine the algebraic and geometric multiplicity of ? = 1. 5. Is the matrix from Question 4 diagonalizable? 6. Let A an n × n symmetric matrix. Show that the transposes of the (right) eigenvectors of A are left eigenvectors of A corresponding to the same eigenvalue.

David · Accepted Answer

1. (a) If det(A) = 1 it tells that m = n.
(b) Aej = cj , jth column. And ε
T
i A = ai, ith row.
(c) εTi Aej = aij
(d)
A.x =

R1
.
.
Rm
 .x =

R1.x
.
.
Rm.x
 =

0
.
.
0

Where Ri is ith row.
(e)
A.x.y = (C1, .., Cn)x.y = (C1x1, ..., Cnxn)y
= (x1C1.y,

Let A ? R m×n , let~ai = (ai1 ai2 ··· ain) be the i th row of A,~cj = ? ???? a1 j a2 j . . . am j ? ???? the j th column of A, let En = {~e1, ~e2, ..., ~en} be the standard basis for R n , and let Em...

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