Question

How do you solve `x + 4y = 8` and `2x + y = 9` using matrices?

Answer 1

`x = 4, y = 1`

Explanation:

In preamble, we can write these equations in column vector form, ie:

`x ((1),(2)) + y ((4),(1)) = ((8),(9))`

And so they can also be written in matrix form, as we then have a set of rules in matrix algebra which means they are the same thing:

`((1, 4),(2, 1)) ((x),(y)) = ((8),(9))`

That is: `M mathbf x = mathbf b`

From here, we have choices. We can row reduce, but as we are looking at the simplest matrix algebra (I think), we should try to use the inverse matrix `M^(-1)`, the idea being:

`M^(-1)M mathbf x = M^(-1) mathbf b`

`implies I mathbf x = M^(-1) mathbf b`

Generally speaking, for a `2 times 2` matrix, `M = ((a,b),(c,d))`, the inverse `M^(-1)` is:

`M^(-1) = 1/(ad - cb) ((d,-b),(-c,a))`.

So we have:

`((1, 0),(0, 1)) ((x),(y)) =- 1/7 ((1, -4),(-2, 1)) ((8),(9))`

`implies ((x),(y)) =- 1/7 ((-28),(-7))`

`implies ((x),(y)) = ((4),(1))`