Solve the system by using matrices (and inverse matrices)

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.

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