## Final Exam, Math. 221 (s. 02), 11 May 2004

Scoring: Each of questions 1-10 is worth twelve points. Partial credit may be earned.

Top score possible is 120 points, with 100 points probably representing a very respectable grade.

In any event, letter grades will be distributed along a suitable "curve."Suggested answers starting to be available now; more details soon.

1. For which value(s) of the constant

cdoes the system of equations

x+y+ 2z=a

x+z=b

2x+y+ 3z=c

have

(i) infinitely many solutions?

(ii) more than one solution, but not infinitely many?

(iii) exactly one solution?

(iv) no solution at all?

Explain your reasoning, and exhibit the solutions that are available.2. Solve the system of equations

3/x+ 2/y− 1/z= −15

5/x+ 3/y+ 2/z= 0

3/x+ 1/y+ 3/z= 11

forx,y, andz, explicitly explaining all relevant strategies and calculations.3. Defend or rebut the following allegation(s).

(i) For alln×nmatricesAandB, Trace(A+/−B) = Trace(A) +/− Trace(B).

(ii) For alln×nmatricesAandB, Trace(AB) = Trace(A)Trace(B).

(iii) For alln×nmatricesAandB, Trace(AB) = Trace(BA).

[Suggestion:for item (iii), what's the relevance of the matrixCwhose (ij)^{th}entryc_{ij}

is the productc=_{ij}aof the (_{ij}b_{ji}ij)^{th}entryaof_{ij}Awith the (ij)^{th}entrybof_{ji}B?^{T}

And can you add up all the entries ofCin different ways, still obtaining the same total sum?]4. Suppose a 5×5 matrix

Mhas one of the forms| * * 0 0 0 | | * 0 0 0 * | | 1 * * * * |

| * * 0 0 0 | | 0 0 * 0 0 | | 0 1 * * * |

A= | * * 0 0 0 | ,B= | 0 * 0 * 0 | ,C= | 0 0 * * * | .

| * * * * * | | 0 0 * 0 0 | | 0 0 0 1 * |

| * * * * * | | * 0 0 0 * | | 0 0 0 0 1 |(i) What are all the values can you obtain for det(

M) (M=A,B,C) by substituting

numerical values (not necessarily all the same) for the *'s? Explain your reasoning.

(ii) Which of these forms can be invertible, and when exactly?

5. What can one say about the behavior of therowsof a square matrixAwhosecolumnsare all mutually perpendicular and of length one? Explain your reasoning.6. A (new) definition: a (square) matrix

M(n×n, say, withn>1) is said to beidempotentifMM=M.

(i) Show that, if a real number λ is an eigenvalue of an idempotent matrix, then either λ = 0 or λ = 1.

(ii) If square matrixAis idempotent, andIis the identity matrix the same size and shape asA,

show thatI−Ais idempotent too.

(iii) Show that when idempotentAis not the zero matrix, then 1 is an eigenvalue forA,

while when idempotentAis not the identity matrix, then 0 is an eigenvalue forA.

(iv) Given column vectorzinR^{n}, letx=Azandy= (I−A)z.

EvaluateAx,Ay, (I−A)x, (I−A)y, andx+y, and determine whether or notxandyare linearly independent.

(v) Discuss any relationships you can now discern among

the column spaces, the eigenspaces, and the nullspaces ofAand ofI−A.

(vi) Establish why every idempotent matrix must be diagonalizable; or, if that's not true, present a counterexample.7. Suppose

n>1 and then×nmatrixMisnilpotent(another new definition: we call

a (square) matrixMnilpotentif some iteratedm-fold product ofMwith itself vanishes:

(†)M^{m}=M · M ·...· M=0(here0stands for the matrix with all zero entries)).

←mfactors →(i) Determine the possible eigenvalues of the nilpotent matrix

M.

(ii) Withk> 0 an arbitrary integer, how do the column spaces ofM^{k}andM^{k+1}compare?

(iii) Explain why, if everM^{k}=M^{k+1}, it will follow thatM^{k}=0.

(iv) Show that, forn×nnilpotentM, an integermas in (†) can always be found satisfyingm<n.8. Consider the following four vectors in

R_{3}:

the vector from the origin to (1, 1, −1), that from the origin to (1, −1, 1),

that from the origin to (−1, 1, 1), and that from the origin to (−1, −1, −1).

(i) What are their lengths? (Needed for part (ii).)

(ii) What are the angles (or, at least, the cosines of the angles) between any two of them?

(NB: there are 6 such angles(!). The four non-(0, 0, 0) points mentioned form the vertices of atetrahedron.)9. By a

zerovectorfor a vector spaceV(with addition +) some people mean any vectorzsatisfying:

for all vectorsvinV,z+v=v=v+z.

By evaluatingz+z' two different ways, show that any two zero vectorszandz' forVmust be equal.10. Describe briefly any two components of this course that you feel this exam has unfairly ignored, despite their being absolutely central to the development of more advanced vector/matrix ideas.

**Reminder.**
**Please indicate explicitly, on what you turn
in, that you have respected Wesleyan’s Honor Code.**

Last revised: 11 May 2004