Question

elementary matrix. I think it has something to do with the column but i thought it had to be row operstions

Question: find an elementary matrix E such that AE=B

i am given A = 2 4
1 6

B= 2 -2
1 3

i set it up like
(2 4|1 0) = (2 -2)
(1 6|0 1) (1 3)

the 2 and 1 are good but i cant do anything to the 4 and 6 without changing the 2 and 1 which i want to leave. Any thoughts?

Answer 1

I have no idea what method you're using but here's a different perspective on it.

You're looking for a matrix E that verifies AE=B.

There's two easy ways to solve this.

Answer 2

I dont know if there is a name for it but basically i have matrix A and am using the identity matrix next to it. i think im supposed to do row operations on the Matrix A and do the same to the identiy matrix until it equals Matrix B

Answer 3

Method 1

First and foremost figure the dimensions of E - since A and B are two 2x2 matrixes then it's pretty clear that E will also be a 2x2 matrix.

Let x1,x2,x3 and x4 be the elements of that 2x2 matrix.

By solving AE and then equating that to each respective value of B you will get a linear system of 4 equations with 4 unknown variables (x1,x2,x3 and x4) which should be pretty easy to solve.

Answer 4

Method 2

This implies knowing about the inverse of matrixes.

We work on the same deduction that E is a 2x2 matrix with elements x1,x2,x3 and x4 but instead of making equations out of it we could use a neat trick with the inverse matrix of A.

Let A^(-1) be the inverse of matrix A.

By multiplying A^(-1) to the "left" side of the equation we have that

A^(-1)*A*E=A^(-1)*B

But as we all know, the inverse of a matrix multiplied with the matrix itself is the identity matrix - which never changes anything about any matrix when multiplied to it.

E=A^(-1)*B

So all you need to do now is to find the inverse of the matrix A (that is A^(-1) ) and multiply it with B and then x1,x2,x3 and x4 will equal each corresponding member of that.

Answer 5

ok, i think thats kind of what i was doing. with the identity matrix. but i was thinking row operations or something. at least thats what my book did. but im going to try it your way.

Answer 6

Does it connect to the Gauss-Jordan elimination method ?

Answer 7

yes

Answer 8

Oh, well - if your teacher really insist on you using that then I suppose you can determine A^(-1) using that method as to prove to him/her that you know the method.

Answer 9

Here are some details.

Hope this helps.