Application of Matrices

Having covered the inverse of a matrix, this section turns to a major application: solving a system of linear equations using matrices. The first step is always to convert the given system of equations into the form of a single matrix equation.

Worked illustration. Consider the two linear equations $2x+3y=5$ and $x-4y=9$. Recalling that a $2\times1$ column matrix equals another $2\times1$ column matrix only when their corresponding entries match, these two equations can be written together as one matrix equality:

Now, using the definition of matrix multiplication, the left-hand column can itself be written as a product:

Naming $\begin{bmatrix}2&3\\1&-4\end{bmatrix}=A$ (the coefficient matrix), $\begin{bmatrix}x\\y\end{bmatrix}=X$ (the column matrix of variables) and $\begin{bmatrix}5\\9\end{bmatrix}=B$ (the column matrix of constants), the whole system collapses to the single equation $AX=B$.

If $A$ is of order $2\times2$ and $X$ is of order $2\times1$, then $B$ is necessarily of order $2\times1$ (matrix multiplication forces this). The same construction extends to three linear equations in three variables $x,y,z$: the system is written $AX=B$ with $A$ now $3\times3$, $X=\begin{bmatrix}x\\y\\z\end{bmatrix}$ of order $3\times1$, and $B$ of order $3\times1$. …