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 A^T : 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 A^T , all

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

(b) have the same row space?     -- Yes -- because that will be all of R_n 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 A − xI_2×2 ;     -- that's

	p − x	q
A − xI2×2     =
	r	s − x

(b) work out the determinant of that matrix A − xI_2×2 ;     -- that's (p − x)(s − x) − qr , which (answering part (c)) becomes x² − (p + s)x + ps − qr .

(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 = ps − qr 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 a_{i j} , and B , with entries b_{i j} .

(a) Work out the three principal diagonal entries of the product AB ;     -- these are
a_{1 1}b_{1 1} + a_{1 2}b_{2 1} + a_{1 3}b_{3 1} ,
a_{2 1}b_{1 2} + a_{2 2}b_{2 2} + a_{2 3}b_{3 2} , and
a_{3 1}b_{1 3} + a_{3 2}b_{2 3} + a_{3 3}b_{3 3} .

(b) work out the three principal diagonal entries of the product BA .     -- these are
b_{1 1}a_{1 1} + b_{1 2}a_{2 1} + b_{1 3}a_{3 1} ,
b_{2 1}a_{1 2} + b_{2 2}a_{2 2} + b_{2 3}a_{3 2} , and
b_{3 1}a_{1 3} + b_{3 2}a_{2 3} + b_{3 3}a_{3 3} .

(c) Using parts (a) and (b), evaluate and compare trace(AB) , trace(BA) , and trace(AB − BA) .
    -- 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 a_{i j}b_{j i} of the one trace match up perfectly with the corresponding summands b_{j i}a_{i 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 AB − BA = I_3×3 ? Why so? or why not? (Hint: what is trace(I_3×3) ?)
    -- No way -- AB − BA always has trace = zero, but trace(I_3×3) = 1 + 1 + 1 = 3 .