Homework Help: Canonical matrix question

1. Jan 3, 2009

franky2727

given A={(1,2,1),(2,4,2),(3,6,3)} findr and invertable matrices Q and P such that Q-1AP={(Ir,0),(0,0)} where each zero denotes a matrix of zeros not necessarily the same size

paying special attension to the order of the vectors write down the bases of R3 with respect to which Q-1AP represents the mapping x->Ax

i think i can do the first part getting row opps of r3-3r1 and r2-2r1 and then column opps of c2-2c1 and c3-c1 giving me {(1,0,0),(0,0,0),(0,0,0) and therefore r=1

then i do I3 with the same row opps giving Q-1={(1,0,0),(-2,1,0),(-3,0,1)) giving Q=(Q-1)-1 = {(1,0,0),(1/2,1,0),(1/3,0,1)

same with the column opps on P gives me {1.-2.-1),(0,1,0),(0,0,1)}=P

i beleive this is right but have not done it in a while and may be messing up the method so a check wouldnt go a miss, also i dont know how to do the second part of the question, it looks slightly familia with the getting vectors in the right order but i cant remember where to start so help here would be aprichiated thanks. on a side note this is revision not homework so feal free to splurt it all out :P

Last edited: Jan 3, 2009
2. Jan 3, 2009

HallsofIvy

It's not difficult to check it yourself is it?

If Q-1 and P are as you say then you want to calculate
$$Q^{-1}AP= \begin{bmatrix}1 & 0 & 0 \\ -2 & 1 & 0 \\ -3 & 0 & 1 \end{bmatrix}\begin{bmatrix}1 & 2 & 1 \\ 2 & 4 & 2\\ 3 & 6 & 3\end{bmatrix}\begin{bmatrix}1 & -2 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$

Is that equal to
$$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$$?

Last edited by a moderator: Jan 3, 2009
3. Jan 3, 2009

franky2727

ah ye, silly me, what about the second part? no ideas where to start there

4. Jan 4, 2009

franky2727

how is this done or even started?

5. Jan 4, 2009

franky2727

what does this " with respect to which Q-1AP represents the mapping x-> Ax mean?