Question

Suppose A is an m by n matrix

Answer 1

If [ A | Im ] is row reduced to a matrix [B | P], explain why P must be a nonsingular matrix, such taht PA = B

Answer 2

Somebody, HELP!!!

Answer 3

*such that

Answer 4

Is that Im thing the mxm identity matrix?

Answer 5

yes

Answer 6

Well, P must be non-singular because it is row equivalent to the identity matrix. As for the rest, hrm..

Answer 7

well here's the solution my book gives, i just don't get it..

Answer 8

one moment, uploading..

Answer 9

There part a

Answer 10

I don't get the explanation.. and I'm not that dumb lol.. by the way I'm bahrom7893.. haha this is my question asking account..

Answer 11

Ok yeah. I was going to suggest elementary row op matrices, but wasn't sure if it'd been covered yet.

Answer 12

You are familiar with elementary row op matrices?

Answer 13

hmm yea.. is it like when you have [A | I] right and then you try to get the A into reduced row echelon form, and with each row operation you make, the I also changes.. so once the I changes, the first step would be the first elementary row op matrix and the next one would be the 2nd row operation applied to the Identity again, but from the beginning..

Answer 14

like if i multiplied row 1 by -2 and added to the 2nd row, my 1st elem matrix would be like:
100
-210
001

Answer 15

im talkin 3x3 here..

Answer 16

And if the next step is multiply say row two by 2 and add to the last row, then elem matrix two would be:
100
010
021

Answer 17

Is that right? anyway ill just let u finish whatever u're typing..

Answer 18

Right. So now imagine you take the product of all those row op matrices:

\[\prod_{n=1}^k E_n = P\]

Since you started with I and multiplied it by each of those row ops to get to P.

Answer 19

okay..

Answer 20

But you also multiplied A by each of those row ops to get B.

Answer 21

yea..that makes sense..

Answer 22

So PA = B because P is just the product of all the row ops you used to get from A to B.

Answer 23

Ohh I sort of get it now... thanks.. can you stay on for a bit longer.. linear algebra is not really my thing.. I am studying for my test on monday.. and the soln manual is not helpin..

Answer 24

\[P = (\prod_{i=1}^nE_i)I = \prod_{i=1}^nE_i\]
\[B = (\prod_{i=1}^nE_i)A = PA\]

Answer 25

but then wait how would i explain this?
P is a product of all elementary matrices that were obtained by reducing A to B, which is why it is a nonsingular matrix and PA = B so that's it?

Answer 26

It's non-singular because it's row equivalent to the identity matrix. PA=B for the reason I stated in English and algebraically.

Answer 27

okay thanks.