Question

How do you solve `6x + 2y = -16` and `5x + 6y = 4` using matrices?

Answer 1

Use the matrix equation and the inverse matrix to find `[(x),(y)]= [(-4),(4)]`

Explanation:

We begin by writing our system of equations in matrix form

`[(6,2), (5,6)][(x),(y)]=[(-16),(4)]`

Let `A=[(6,2), (5,6)]`

We then use the identity that a matrix, `A` multiplied by its inverse, `A^(-1)` is the identity matrix, `I`, i.e.

`A^(-1)A=A A^(-1)=I`

Multiplying both sides of our original matrix equation by the inverse matrix we get:

`A^(-1)A[(x),(y)]=A^(-1)[(-16),(4)]`

which simplifies to

`[(x),(y)]=A^(-1)[(-16),(4)]`

since any matrix or vector multiplied by the identity matrix is itself. We now need to find `A^(-1)`. Use the method described here How do I find the inverse of a `2xx2` matrix? to find the inverse to be:

`A^(-1)=[(3/13,-1/13), (-5/26,3/13)]`

Substitute this into the equation to find `x` and `y`

`[(x),(y)]=[(3/13,-1/13), (-5/26,3/13)][(-16),(4)] = [(-4),(4)]`

Therefore `x=-4` and `y=4`