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 R^{n} 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 − xI_{2×2} = | ||||
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 x^{2} − (p + s)x + ps − qr .
(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) ; -- it's clearly the constant term c_{0} = 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 c_{1} = −(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 x^{2} ) 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 .