Math. 221 (s. 02), Test Two, 20 April 2004
Also available: a brief answer-sheet to this exam.
Scoring: Each lettered part of each of problems 1-5 is worth four points. Partial credit may be earned. Top score possible is 100 points. Letter grades will be distributed along a suitable "curve."
Time allowed: 40 minutes. (Note that most of these questions take longer to state than to answer.)
Please indicate explicitly, on what you turn
in, that you have respected Wesleyan’s Honor Code.
1. Must a matrix A and its fully reduced row-echelon form frref(A)
(a) have the same rows?
(b) have the same row space?
(c) have the same columns?
(d) have the same column space?
(e) have the same null space?
2. Same five sub-questions, but this time about a matrix A and its transpose A^{T} : must they
(a) have the same rows?
(b) have the same row space?
(c) have the same columns?
(d) have the same column space?
(e) have the same null space?
3. Suppose now that the given matrix A is (square and) invertible: must A , its fully reduced row-echelon form frref(A) , and its transpose A^{T} , all
(a) have the same rows?
(b) have the same row space?
(c) have the same columns?
(d) have the same column space?
(e) have the same null space?
4. Given a real number x and a little 2-by-2 matrix A
p | q | |||
A = | ||||
r | s |
(a) Write out the matrix A − xI_{2×2} ;
(b) work out the determinant of that matrix A − xI_{2×2} ;
(c) simplify that determinant to an expression of the form c_{2}x^{2} + c_{1}x + c_{0} , for suitable coefficients c_{0} , c_{1} , and c_{2} ;
(d) tell which of those coefficients, if any, is somehow related to det(A) ;
(e) tell which of those coefficients, if any, is somehow related to trace(A) (the trace of A , remember, is the sum of the entries on the principal diagonal of A) ;
(f) tell what, if anything, can be said about the remaining coefficient(s).
5. Suppose given two 3-by-3 matrices A , with entries a_{i j} , and B , with entries b_{i j} .
(a) Work out the three principal diagonal entries of the product AB ; and
(b) work out the three principal diagonal entries of the product BA .
(c) Using parts (a) and (b), evaluate and compare trace(AB) , trace(BA) , and trace(AB − BA) .
(d) Can there be 3-by-3 matrices A , B , for which AB − BA = I_{3×3} ? Why so? or why not? (Hint: what is trace(I_{3×3}) ?)
Please remember
to indicate explicitly on what you turn
in that you have respected Wesleyan’s Honor Code.