Matrices

You can represent a system of linear equations as a matrix by removing the algebraic letters and using only the coefficients.

For example, the system

\[ x + 2y + 3z = 18 \] \[ 3x + 4y + z = 12 \] \[ 2x - y + z = 6 \]

can be represented as

\[ \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 1 \\ 2 & -1 & 1 \\ \end{bmatrix} = \begin{bmatrix} 18 \\ 12 \\ 6 \end{bmatrix} \]

The matrix after the \( = \) equality is the right hand side matrix which is shortened to \( rhs \).

Implictly there is also a matrix containing the unknowns by which each coefficient in a row is multipled.

\[ \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 1 \\ 2 & -1 & 1 \\ \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 18 \\ 12 \\ 6 \end{bmatrix} \]

So if we multiply \( x \) by the \( 1 \), \( 2 \) and \( 3 \) values in the first row we will get back to the first equation in the system.

\[ x + 2x + 3z = 18 \]

You can represent the entire system including the solution in a single matrix by removing the implicit \( = \) equality sign.

\[ \begin{bmatrix} 1 & 2 & 3 & 18 \\ 3 & 4 & 1 & 12 \\ 2 & -1 & 1 & 6 \\ \end{bmatrix} \]

Ax Form

You can express a the matrix form of a system of linear equations as another equation

\[ Ax = b \]

Where

• \( A \) is the matrix of coefficients
• \( x \) is the unknowns, and
• \( b \) is the solution

There is a value that \( b \) can take such that there is always a solution for \( x \) no matter what \( A \) is. This is

\[ Ax = 0 \]

Where every element in \( b \) is \( 0 \) then the matrix solution can satisfied by setting every element of \( x \) to \( 0 \) as well so that when the coefficients are multiplied they also equal \( 0 \).

\[ \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 1 \\ 2 & -1 & 1 \\ \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]

Matrix operations

You can multiply or divide every element in a matrix row to manipulate the coefficients and the solution easily. For example we can turn \( x + 2x + 3z = 18 \) into \( 2x + 4x + 6z = 36 \) easily by doing

\[ \begin{bmatrix} 1 & 2 & 3 & 18 \end{bmatrix} \cdot \begin{bmatrix} 2 \end{bmatrix} = \begin{bmatrix} 2 & 4 & 6 & 36 \end{bmatrix} \]

You can subtract or add matrix rows to one another.

Row-Echelon Form

A row-echelon form matrix has

• Every non-zero number is not brlow a row containing a \( 0 \)
• Every row’s pivot - first non zero number - is right of the pivot in the row above

\[ \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 0 & 0 & 6 & 7 & 8 \\ 0 & 0 & 0 & 9 & 10 \end{bmatrix} \]

Is an example of a valid row-echelon matrix.