## anonymous 4 years ago obtain the row rank and parametrically represented solution of the system. 3x-y+3z=5 x+2y+2z-3w=-1 2x+5y+4z+2w=10

1. amistre64

augment the coeff matrix

2. amistre64

row rank = colA = dimA = number of pivot points

3. anonymous

i didn't understand any of that.

4. amistre64

then you really need to review your material, or ask more probing questions

5. anonymous

what exactly is row rank?

6. amistre64

make sure each equation HAS the same variables; whatevers missing put a 0 to it so that it is included 3x -1y+3z+0w=5 1x+2y+2z-3w=-1 2x+5y+4z+2w=10 strip off everything thats NOT a number 3 -1 3 0 5 1 2 2 -3 -1 2 5 4 2 10 now this is a coeff matrix that you can augment

7. amistre64

row rank is the number of nonzero rows that you end up with

8. anonymous

i was about to say this........

9. amistre64

if we format this for the wolf, life gets a bit easier :) rref{{3, -1, 3, 0, 5},{1, 2, 2, -3, -1},{2, 5, 4, 2, 10}}

10. anonymous

ok i got that. what's next?

11. amistre64
12. amistre64

there are 3 nonzero rows; so row rank = 3 the last column of the augmented matrix is a constant vector the next to last is the a free variable column

13. amistre64

our parametric is based on these last 2 columns in this case; our constant vector and our free vectors

14. amistre64

$\vec x=\begin{pmatrix}x\\y\\z\\w \end{pmatrix}=\begin{pmatrix}109/3\\12\\-92/3\\0\end{pmatrix}+w\begin{pmatrix}-73/3\\-8\\65/3\\1 \end{pmatrix}$

15. anonymous

in this typa of questions, i was taught to suppose an arbitary constant and then calculate values of x,y,z,w for a particular value of arbitary constant.

16. amistre64

yes, in this case the free variable is what you are calling an arbitrary constant

17. amistre64

the augmented matrix produces our vectors for us; by augmenting with the "=" column vector to begin with we produce a particular value vector; and all thos coumns that dont stand on a "1" are called free variables ... arbitrary constants

18. amistre64

it means that the rest of it can be defined using just those column vectors