2.2 Matrices

Section 1 — Writing Simultaneous Equations as a Matrix Equation

What is a system of linear simultaneous equations?

A linear simultaneous equation is simply an equation involving unknowns (like $x$, $y$, $z$) where none of them are squared or multiplied together — just plain multiples of each variable added together. A "system" means you have several of these equations at the same time, and you want to find values of $x$, $y$, $z$ that satisfy all of them at once.

For example, consider this system of 3 equations in 3 unknowns:

$2x + y - z = 5$ $x - 3y + 2z = 4$ $3x + 2y + z = 1$

Writing it as a matrix equation

A matrix equation takes the form:

$A\mathbf{x} = \mathbf{b}$

Where:

• $A$ is a 3×3 matrix made from the coefficients (the numbers in front of $x$, $y$, $z$)
• $\mathbf{x}$ is a column vector of the unknowns
• $\mathbf{b}$ is a column vector of the right-hand side numbers

For the example above:

$A = \begin{pmatrix} 2 & 1 & -1 \\ 1 & -3 & 2 \\ 3 & 2 & 1 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 4 \\ 1 \end{pmatrix}$

So the full matrix equation is:

$\begin{pmatrix} 2 & 1 & -1 \\ 1 & -3 & 2 \\ 3 & 2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \\ 1 \end{pmatrix}$

You can check this works by multiplying out — each row of $A$ times $\mathbf{x}$ gives one of the original equations.