System of Linear Equations

Any system of linear equations can be represented in matrix notation as shown below.

• $a_{11}x+a_{12}y+a_{13}z=b_1$
• $a_{21}x+a_{22}y+a_{23}z=b_2$
• $a_{31}x+a_{32}y+a_{33}z=b_3$

$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} b_{1} \\ b_{2} \\ b_{3} \\ \end{pmatrix} \implies AX=B$$

2 types based on $B$: $$

• $=0$: Homogeneous system
• $\neq0$: Non-homogeneous system

Number of solutions

A system of equations can have 0 or 1 or infinitely many solutions.

Consistent

When the system of equations has at least 1 solution.

Inconsistent

When the system of equations has no solution.