An extremely useful application of matrices is for solving systems of linear equations. Consider the following system of equations in its algebraic form.

$\begin{aligned}
a_1 x + b_1 y + c_1 z = d_1 \\
a_2 x + b_2 y + c_2 z = d_2 \\
a_3 x + b_3 y + c_3 z = d_3
\end{aligned}$

This system of equations can be represented using vectors and their linear combination operations that we discussed in the previous exercise. We combine the coefficients of the same unknown variables, as well as the equation solutions, into vectors. These vectors are then scalar multiplied by their unknown variable and summed.

$x 
\begin{bmatrix}
a_1 \\
a_2 \\
a_3 \\
\end{bmatrix}

+
y
\begin{bmatrix}
b_1 \\
b_2 \\
b_3 \\
\end{bmatrix}
+
z
\begin{bmatrix}
c_1 \\
c_2 \\
c_3 \\
\end{bmatrix}
=
\begin{bmatrix}
d_1 \\
d_2 \\
d_3 \\
\end{bmatrix}$

Our final goal is going to be to represent this system in form Ax = b, using matrices and vectors. As we learned earlier, we can combine vectors to create a matrix, which we’ll do with the coefficient vectors to create matrix A. We can also convert the unknown variables into vector x. We end up with the following Ax = b form:

$\begin{bmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3 \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}
=
\begin{bmatrix}
d_1 \\
d_2 \\
d_3 \\
\end{bmatrix}$

Click below if you would like to see how these two expressions are equivalent:

$\begin{bmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3 \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix} =
x 
\begin{bmatrix}
a_1 \\
a_2 \\
a_3 \\
\end{bmatrix}

+
y
\begin{bmatrix}
b_1 \\
b_2 \\
b_3 \\
\end{bmatrix}
+
z
\begin{bmatrix}
c_1 \\
c_2 \\
c_3 \\
\end{bmatrix}$

Click to see proof

We can also write Ax = b in the following form:

$Ax = b
\rightarrow
\begin{bmatrix}
A  \bigm|  b
\end{bmatrix}$

This means we can write our Ax = b equation above as:

$\begin{bmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3 \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}
=
\begin{bmatrix}
d_1 \\
d_2 \\
d_3 \\
\end{bmatrix}

\rightarrow

\begin{bmatrix}
    a_{1} & b_{1} & c_{1} \bigm| d_{1} \\
    a_{2} & b_{2} & c_{2} \bigm| d_{2} \\
    a_{3} & b_{3} & c_{3} \bigm| d_{3} \\ 
\end{bmatrix}$

This is what is known as an augmented matrix, and we will use these in the following exercises to solve for linear systems!