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A math exam for a linear algebra and economics course (math 205b). The exam covers topics such as finding a basis for the column space of a matrix, determining necessary and sufficient conditions for a vector to be in the column space, and calculating the determinant and eigenvalues of a matrix. It also includes problems related to an economic model of a four-sector economy.
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; then [^ A^ |^ I^4 ] is row-equivalent to
Let R = rref(A).
1A. Find a basis for Col(A).
1B. Find a basis for Col(R).
1C. Is 0 an eigenvalue of A? If so, find a basis for the eigenspace of 0, if 0 is not an eigenvalue, explain why not. (You do not need to find any polynomials in this problem!)
This is a continuation of problem 1. For your reference we copy the following:
; and [^ A^ |^ I^4 ] is row-equivalent to
1D. Find all conditions both necessary and sufficient for a vector b with entries b 1 ,... , b 4 to be in Col(A).
1E. Find all conditions both necessary and sufficient for a vector b with entries b 1 ,... , b 4 to be in Col(R).
1F. Find a non-zero vector which is in Col(A) but not Col(R). Show your reasoning.
1G. Find a non-zero vector which is in Col(R) but not Col(A). Show your reasoning.
1H. Find a non-zero vector which is in both Col(A) and Col(R). Show your reasoning.
Math 205B Exam 02 page 4. 03/19/2010 Name
3A. Find the characteristic polynomial of A. Show all your work. Choose your “expansion” wisely and don’t make any sign mistakes at all. Show all your work. Write your polynomial in completely factored form.
3B. What are the eigenvalues of A, and their multiplicities?
3C. Find a basis for the eigenspaces of each of those eigenvalues. Organize your work nicely; make sure it’s clear which basis goes with which eigenvector.
3D. Is A diagonalizable? If so find P , D and P −^1 with the appropriate properties; if A is not diagonalizable, explain why it’s not.
4A. Show that B has the same characteristic polynomial as does the matrix A in problem 3. Show your work, make no sign errors, and again, write your polynomial in completely factored form.
4B. This matrix B is not diagonalizable. Given this, what must be the case about one of the eigenvalues and the dimension of its eigenspace? (In your answer, say which eigenvalue has to be “at fault”).
4C. Verify your answer to 4C by finding that eigenspace.
4D. Even though B is not diagonalizable, you should still be able to find a matrix Q such that BQ = QD where D is the same as in problem 3. Do it.
x y z 3 4 6 a b c
(^) has determinant 10.
Find the determinant of each of the following matrices. You do NOT need to list any rules about matrices you used to find the det. (eg, “swapping rows changes the sign of the det” or “the inverse of the derivative of a matrix is the matrix of its eigenzeigen” (this second fact is nonsense). Just FIND the determinants.
x − 18 y − 24 z − 36 3 4 6 a b c
(^) the det is: 6B.
a b c
(^) the det is:
x y z 1 / 4 1 / 3 1 / 2 a b c
(^) the det is: 6D.
x 3 a y 4 b z 6 c
(^) the det is:
6E. 4 B the det is: 6F. B−^1
BONUS: At the review we mentioned that the intersection of two subspaces of a vector space is another subspace. Find a basis for Col(A)∩ Col(R) for A and R from problem 1. Use the back of this sheet if necessary.
