How do you solve simultaneous equations using matrices?

Answer 1

There are many ways of doing this. For example Gaussian elimination, diagonalising, but the simplest to explain is matrix inversion (I'm assuming some knowledge of matrices here, and unfortunately some of the matrix formatting is a little off due to limitations in the editor):

Any system of simultaneous equations can be rewritten as the matrix equation

A.v = u

The coefficients of the variables become the entries in the square matrix, A.

To solve the matrix equation we need to invert A, and then multiply by the inverse, giving us

I.v = A-1.u

where I is the identity matrix.

As an example take the following system of equations:

2x - 3y = 1

4x - 5y = 5

The matrix version of this equation is

{ 2 -3 } { x } = { 1 }

{ 4 -5 } { y } { 5 }

A

v

u

It's clear that if you multiply out the matrix row by row, you get the original set of equations.

In our case

I = { 1 0 }

{ 0 1 }

A-1 = { -2.5 1.5 }

{ -2 1 }

(Finding the inverse of a matrix is a whole other question)

so A-1.u = { 5 }

{ 3 }

Therefore we have x = 5, and y = 3.

Inversion of A is the most difficult step, though this can easily be done with a computer.