Q: Solve the system of equations by using matrices (and inverse matrices)
7x + 5y = 3
3x – 2y = 22
Answer:
Okay, first set up the coefficient matrix (A), the variable matrix (X) and the “answer matrix” (B) so that:
A*X = B
| 7 5 | | x | = | 3 |
|3 -2| | y | |22|
A X = B
So, if A*X = B, and we want to solve for X, we know with matrices that:
A^-1 * A*X = A^-1 * B (Note: A^-1 means A inverse)
Which simplifies to:
X = A^-1 * B
So, all we need to do is find A^-1 and multiply it by B to solve for X.
To find an inverse of a 2 x 2 matrix:
If A =
| a b |
| c d |
then A^-1 =
1/(detA) *
| d -b |
| -c a |
So, in our case:
A =
| 7 5 |
|3 -2|
And the det(A) = -14 – 15 = -29
So,
A^-1 =
(1/-29) *
| -2 -5 |
| -3 7|
A^-1 =
| 2/29 5/29 |
|3/29 -7/29 |
Now take A^-1 and multiply it by matrix B… You should get (didn’t show work):
A^-1*B =
| 4 |
| -5|
So, x = 4 and y = -5.