# 04 Solving Systems of Equations Using Matrices

Before we begin, we need to understand the **augmented matrix.**

# Augmented Matrix

An augmented matrix is a type of matrix that can be used to solve systems of equations.

Given a system of linear equations:

$$ \begin{aligned} &a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n} =b_{1} \\ &\vdots \\ &a_{m_{1}}x_{1}+a_{m2}x_{2}+\cdots+a_{m}x_{n}=b_{m} \end{aligned} $$We can extract the coefficients ($a$’s) into a matrix:

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots &\ddots & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$Now include the other side of the equation using an augmented matrix:

$$ \left[\begin{array}{ccc|c} a_{11} & \cdots & a_{1n} & b_{1} \\ a_{21} & \cdots & a_{2n} & b_{2} \\ \vdots \\ a_{m1} &\cdots & a_{mn} & b_{m} \end{array}\right] $$The bar is like an = sign.

# Solving a System

- Given a system of linear equations:

$$ \begin{aligned} &a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n} =b_{1} \\ &\vdots \\ &a_{m_{1}}x_{1}+a_{m2}x_{2}+\cdots+a_{m}x_{n}=b_{m} \end{aligned} $$ Convert the system into an augmented matrix.

- We can now use elementary operations to get our augmented matrix into reduced row echelon form (RREF) to get our solutions.

Tips:

- If there is a pivot outside the coefficient matrix, it means that there are no solutions to the system (the system is inconsistent).
- If the system is consistent, there are two possibilities:
- If there are any free variables, there are infinitely many solutions.
- If there are no free variables (every column has a pivot), there is one unique solution which can be solved for.
- A row of zeros does not imply anything.

## Example Problem

Solve for $x, y,$ and $z$ in:

$$ \begin{aligned} -4x+3y +3z&=-2 \\ -2x+ y &=-3 \\ 3x-2 y -z&=3 \end{aligned} $$- Rewrite the system as an augmented matrix.

- Row reduce the matrix.