Question

How to solve X ? `((-5,5),(-2,5))*X = ((4,5),(5,-1))`

Answer 1

To solve `AX = B`, where `A` and `B` are matrices, we instead solve `X = B * A^-1`.

Explanation:

You might be tempted to try to divide the matrices, e.g. `X = B/A` where `B = ((4,5),(5,-1))` and `A=((-5,5),(-2,5))`. But actually, whenever you want to divide matrices, you instead turn the problem into matrix multiplication. In particular, to solve `AX = B`, we instead solve `X = B * A^-1`.

So first we compute `A^-1`. The formula for the inverse of a `2x2` matrix is:

`((a,b),(c,d))^-1 = 1/(ad-bc)((d,-b),(-c,a))`

That gives us:

`((-5,5),(-2,5))^-1 = 1/((-5*5)-(5*-2))((5,-5),(2,-5))`
`=-1/15((5,-5),(2,-5))`
`=((-1/3,1/3),(-2/15,1/3))`

Now we multiply `B * A^-1`:

`((4,5),(5,-1)) * ((-1/3,1/3),(-2/15,1/3))`
`=((-2,3),(-23/15, 4/3))`