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 ]Find elementary matrices E and F such that EFB = A.
to start off with...do you know how to interchange rows using elementary matrices? ie do you know how to switch row 1 and row 2 using a matrix F?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 ]But there's that third row. What do I do for that?
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 = 10Solve for the values of "a" and "b", and thus for E.
that is not the system you will get...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 = 10Solve for the values of "a" and "b", and thus for E.
Check my work!
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
Oops! You're right: I typoed the multiplication. The system should be as follows:that is not the system you will get...
-a + 9b = 3 2a + b + 10 = 4Where was my head at?