Matrix Forms
November 2nd, 2009
Matrix Forms
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How to solve each system using the inverse of the coefficient matrix?
These are the problems:
1.) 2x-3y=-10
3x+4y -12
2.) 5x-3y=0
10x+6y= -4
Can someone show me the steps to solving this and the answers? I don't remember if I just put them in matrix form and then put a little ^-1 on the outside by using a calculator and multiplying them by -1 or what. The whole process is Greek to me. Thanks!
I will do the first one for you as the second one is similar.
I am assuming that the first one reads:
{ 2x - 3y = -10
{ 3x + 4y = 12.
You are correct that you are supposed to re-write the system into matrix form. Doing so gives:
[ 2 -3 ] [ x ] = [ -10 ]
[ 3 4 ] [ y ]. . .[ 12 ]
Let's say we had matrices A, X, and B, with X being an invertible matrix and:
AX = B.
Multiplying both sides by X^(-1) gives:
AXX^(-1) = X^(-1)B
==> AI = X^(-1)B, since XX^(-1) = I, the identity matrix
==> A = X^(-1)B, since AI = A.
From this point, A would just be the 2x1 matrix on the LHS that contains x and y. Thus, we can just compare the elements of A and X^(-1)B (since they are both equal) to obtain x and y.
So we have:
[ 2 -3 ] [ x ] = [ -10 ]
[ 3 4 ] [ y ]. . .[ 12 ]
(In the form AX = B)
<==>
. . . . .-1 . . . . . . . . . . . . . . . . . . . . . .-1
[ 2 -3 ] * . . .[ 2 -3 ] * . . . [ x ] = [ 2 -3 ] * . . .[ -10 ]
[ 3 4 ]. . . . .[ 3 4 ]. . . . . [ y ]. . [ 3 4 ]. . . . . [ 12 ]
(AXX^(-1) = X^(-1)B)
<==>
. . . . . . . . . . . . . . . .-1
[ 1 0 ] * [ x ] = [ 2 -3 ]. . .*. . . .[ -10 ]
[ 0 1 ]. .[ y ]. . [ 3 4 ]. . . . . . . [ 12 ]
(AI = X^(-1)B)
<==>
[ x ] = [ -4/17 ]
[ y ]. . [ 54/17 ]
(A = X^(-1)B)
Comparing elements yields x = -4/17 and y = 54/17.
I hope this helps!
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