A = [-1 2 ] B = [ 9 1 ]
[ 9 1 ] [-1 2 ]
[ 3 4 ] [ 0 10 ]
Find elementary matrices E and F such that EFB = A.
I can't think how to get started other than just guessing which probably won't work. A hint would be great. Thank you.
A = [-1 2 ] B = [ 9 1 ]
[ 9 1 ] [-1 2 ]
[ 3 4 ] [ 0 10 ]
lawrence wrote:Suppose the matrices A and B are as follows:A = [-1 2 ] B = [ 9 1 ]
[ 9 1 ] [-1 2 ]
[ 3 4 ] [ 0 10 ]
Find elementary matrices E and F such that EFB = A.
I can't think how to get started other than just guessing which probably won't work. A hint would be great. Thank you.
A = [-1 2 ] B = [ 9 1 ]
[ 9 1 ] [-1 2 ]
F = [ 0 1 ]
[ 1 0 ]
FA = [ 0 1 ][-1 2 ]
[ 1 0 ][ 9 1 ]
= [ 0+9 0+1 ]
[ -1+0 2+0 ]
= [ 9 1 ] = B
[-1 2 ]
F = [ 0 1 0 ]
[ 1 0 0 ]
[ 0 0 1 ]
FB = [-1 2 ]
[ 9 1 ]
[ 0 10 ]
E(FB) = [ 1 0 0 ][-1 2 ]
[ 0 1 0 ][ 9 1 ]
[ a b 1 ][ 0 10 ]
-a + 9b = 3
2a + 3b = 10
stapel_eliz wrote:You'll need to multiply B by a 3-by-3 matrix (being the product of square matrices E and F). I think you can use the 2-by-2 you've provided as part of the 3-by-3 version of F that you need.F = [ 0 1 0 ]
[ 1 0 0 ]
[ 0 0 1 ]
FB = [-1 2 ]
[ 9 1 ]
[ 0 10 ]
Now you need to find a 3-by-3 matrix E so that EFB equals A. Since the last row of A contains 3 and 4, you need a matrix which is provide the row-operation(s) necessary to create this from what you've already got (if I'm understanding the process correctly).E(FB) = [ 1 0 0 ][-1 2 ]
[ 0 1 0 ][ 9 1 ]
[ a b 1 ][ 0 10 ]
When you multiply this out, you should get a system of equations:-a + 9b = 3
2a + 3b = 10
Solve for the values of "a" and "b", and thus for E.
Check my work!
stapel_eliz wrote:E(FB) = [ 1 0 0 ][-1 2 ]
[ 0 1 0 ][ 9 1 ]
[ a b 1 ][ 0 10 ]
When you multiply this out, you should get a system of equations:-a + 9b = 3
2a + 3b = 10
Martingale wrote:that is not the system you will get...
-a + 9b = 3
2a + b + 10 = 4