Suggested answers, Math. 221 (s. 02), Test Two, 4/20/2004

1. Must a matrix A and its fully reduced row-echelon form frref(A)

(a) have the same rows?     -- Surely not!

(b) have the same row space?     -- Absolutely!

(c) have the same columns?     -- Surely not!

(d) have the same column space?     -- Nope.

(e) have the same null space?     -- Yes.

2. Same five sub-questions, but this time about a matrix A and its transpose AT : must they

(a) have the same rows?     -- No (not unless A is self-adjoint).

(b) have the same row space?     -- Again, no, not necessarily.

(c) have the same columns?     -- Yet again, no, not necessarily.

(d) have the same column space?     -- Once again, no.

(e) have the same null space?     -- And for the last time, no.

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 AT , all

(a) have the same rows?     -- No, not necessarily.

(b) have the same row space?     -- Yes -- because that will be all of Rn for all of them.

(c) have the same columns?     -- No, not necessarily.

(d) have the same column space?     -- Yes -- because that will be all of Rn for all of them.

(e) have the same null space?     -- Yes -- because that will be {0} for all of them.

4. Given a real number x and a little 2-by-2 matrix A

 p q A     = r s

(a) Write out the matrix AxI2×2 ;     -- that's

 p − x q A − xI2×2     = r s − x

(b) work out the determinant of that matrix AxI2×2 ;     -- that's (px)(sx) − qr , which (answering part (c)) becomes x2 − (p + s)x + psqr .

(c) simplify that determinant to an expression of the form c2x2 + c1x + c0 , for suitable coefficients c0 , c1 , and c2 ;

(d) tell which of those coefficients, if any, is somehow related to det(A) ;     -- it's clearly the constant term c0 = psqr that turns out to be 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) ;
-- it's clearly the coefficient c1 = −(p + s) of x that's related to (is the negative of) trace(A) .

(f) tell what, if anything, can be said about the remaining coefficient(s).     -- the only remaining coefficient (of x2 ) is 1 .

5. Suppose given two 3-by-3 matrices A , with entries ai j , and B , with entries bi j .

(a) Work out the three principal diagonal entries of the product AB ;     -- these are
a1 1b1 1 + a1 2b2 1 + a1 3b3 1 ,
a2 1b1 2 + a2 2b2 2 + a2 3b3 2 , and
a3 1b1 3 + a3 2b2 3 + a3 3b3 3 .

(b) work out the three principal diagonal entries of the product BA .     -- these are
b1 1a1 1 + b1 2a2 1 + b1 3a3 1 ,
b2 1a1 2 + b2 2a2 2 + b2 3a3 2 , and
b3 1a1 3 + b3 2a2 3 + b3 3a3 3 .

(c) Using parts (a) and (b), evaluate and compare trace(AB) , trace(BA) , and trace(ABBA) .
-- The first trace is the sum of all the pieces of answer (a); the second trace is the sum of all the pieces of answer (b); as the various summands ai jbj i of the one trace match up perfectly with the corresponding summands bj iai j of the other (check this!), those two traces (of AB and of BA) are equal; finally, the trace of the difference between AB and BA is the difference trace(AB) − trace(BA) between the traces, which is, of course, zero.

(d) Can there be 3-by-3 matrices A , B , for which ABBA = I3×3 ? Why so? or why not? (Hint: what is trace(I3×3) ?)
-- No way -- ABBA always has trace = zero, but trace(I3×3) = 1 + 1 + 1 = 3 .