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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "MAT2705-01/04 05s Quiz 8\n
Use technology for det, solve, rref." }}}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 5 "setup" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "By a random \+
matrix generator with entries from -3..3, bob repeats until he gets a \+
matrix of eigenvectors which leads to integer entries for A that are n
ot too big in value, but with eigenvalues chosen to be 2,3,3." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "B := matrix([[-1, 0, -1], [-
3, 1, -3], [1, -3, 2]]);\nBinv:=inverse(B);" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A:=evalm(B&*diag(
2,3,3)&*B_inv);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvec
ts(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "nah, let's make smaller
numbers..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "B := matrix
([[-2, -3, 3], [0, 2, 0], [-2, 0, 2]]);\nBinv:=inverse(%);\nA:=evalm(B
&*diag(2,3,3)&*B_inv);\neigenvects(A);" }}}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 46 "A:=matrix([[5, 3, -3], [0, 3, 0], [2, 3, 0]]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 627 "a) Find the eigenvalues and eigen
vectors of the matrix A following the full procedure: evaluate the cha
racteristic equation det(A-lambda I_3) = 0, solve it for the eigenvalu
es, then backsubstitute each into the linear system of equations whose
solution leads to an eigenbasis for the eigenspace associated with th
at eigenvalue, scaling up your basis vectors so they have integer comp
onents. You need to report the characteristic equation, its solutions,
and for each eigenvalue the starting augmented matrix and its rref fo
rm, from which by hand you must solve for the solution space basis. La
bel the basis vectors E1, E2, E3. " }}{PARA 0 "" 0 "" {TEXT -1 83 "b) \+
Finally augment them into a matrix B. Do your results agree with Maple
? Explain." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "response" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 15 "alias(ID=&*()):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 50 "evalm(A-lambda*ID);\ndet(%)=0;\nfactor(%);\nsolve(%);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalm(A-2*ID);\nrref(%)
;\naugment(%,[0,0,0]);\nbacksub(%);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalm(A
-3*ID);\nrref(%);\naugment(%,[0,0,0]);\nbacksub(%);" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 39 "E1:=[1,0,1];\nE2:=[3,0,2];\nE3:=[-3,2,0];
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B:=augment(E1,E2,E3);\ninver
se(B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(inverse(B)&
*A&*B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 227 "Maple does not bother
to double these vectors and it seems to always put E3 before E2 but r
andomly lists the two eigenvalues. The reverse ordering is because Map
le assigns parameters right to left, while we do it left to right." }}
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