SOLUTION: Inverse matrix of 2X+3y=5 4X-2y=7

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Question 1055150: Inverse matrix of 2X+3y=5
4X-2y=7
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Inverse matrix of the coefficient matrix is what I think you're asking for.
A=%28matrix%282%2C2%2C%0D%0A2%2C3%2C%0D%0A4%2C-2%29%29
For a 2x2 system,
B=%28matrix%282%2C2%2C%0D%0Aa%2Cb%2Cc%2Cd%29%29
the inverse matrix is,
B%5E%28-1%29=%281%2FD%29%2A%28matrix%282%2C2%2C%0D%0Ad%2C-b%2C-c%2Ca%29%29
where the determinant, D=1%2F%28ad-bc%29
So then,
D=1%2F%282%28-2%29-4%28-3%29%29%29=1%2F%28-4-12%29=-1%2F16
and
A%5E%28-1%29=%28-1%2F16%29%2A%28matrix%282%2C2%2C-2%2C-3%2C-4%2C2%29%29
A%5E%28-1%29=%28matrix%282%2C2%2C%0D%0A1%2F8%2C3%2F16%2C1%2F4%2C-1%2F8%29%29

Answer by ikleyn(52851) About Me  (Show Source):
You can put this solution on YOUR website!
.
I corrected here minor deficiencies of the previous solution.
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Inverse matrix of the coefficient matrix is
A=%28matrix%282%2C2%2C%0D%0A2%2C3%2C%0D%0A4%2C-2%29%29
For a 2x2 system,
B=%28matrix%282%2C2%2C%0D%0Aa%2Cb%2Cc%2Cd%29%29
the inverse matrix is,
B%5E%28-1%29=%281%2FD%29%2A%28matrix%282%2C2%2C%0D%0Ad%2C-b%2C-c%2Ca%29%29
where the determinant, D = ad-bc%29
So then,
D = %282%28-2%29-4%28-3%29%29%29 = %28-4-12%29 = -16
and
A%5E%28-1%29=%28-1%2F16%29%2A%28matrix%282%2C2%2C-2%2C-3%2C-4%2C2%29%29
A%5E%28-1%29=%28matrix%282%2C2%2C%0D%0A1%2F8%2C3%2F16%2C1%2F4%2C-1%2F8%29%29