Augmented Matrix

An alternate way to write System Of Equations where:

• Coefficients matrix are on the left
• Expected constant matrix is on the right side

$$\left[\begin{array}{cccc}{a}_{11}& a_{12}\dots & a_{1k}& b_1\\ a_{21}& a_{22}\dots & a_{2k}& b_2\\ & & & \\ \dots & & & \\ a_{n1}& a_{n2}\dots & a_{nk}& b_n\\ & & & \end{array}\right]$$

Augmented Subroutine Row Operations

$\lambda R_j\to R_j$ (Multiplying by non-zero constant)

$$\left[\begin{array}{ccccc}{x}_{11}& x_{12}& \dots & x_{1n}& b_1\\ & & & & \\ ⋮& & & & \\ x_{j1}& x_{j2}& \dots & x_{jn}& b_j\\ & & & & \\ ⋮& & & & \\ x_{k1}& x_{k2}& \dots & x_{kn}& b_k\\ & & & & \end{array}\right]\stackrel{\lambda R_j\to R_j}{\to }\left[\begin{array}{ccccc}{x}_{11}& x_{12}& \dots & x_{1n}& b_1\\ & & & & \\ ⋮& & & & \\ \lambda x_{j1}& \lambda x_{j2}& \dots & \lambda x_{jn}& \lambda b_j\\ & & & & \\ ⋮& & & & \\ x_{k1}& x_{k2}& \dots & x_{kn}& b_k\\ & & & & \end{array}\right]$$

$\lambda R_i+R_j\to R_j$ (Adding a multiple of another row)

$$\left[\begin{array}{ccccc}{x}_{11}& x_{12}& \dots & x_{1n}& b_1\\ & & & & \\ ⋮& & & & \\ x_{j1}& x_{j2}& \dots & x_{jn}& b_j\\ & & & & \\ ⋮& & & & \\ x_{i1}& x_{i2}& \dots & x_{in}& b_i\\ & & & & \\ ⋮& & & & \\ x_{k1}& x_{k2}& \dots & x_{kn}& b_k\\ & & & & \end{array}\right]\stackrel{\lambda R_i+R_j\to R_j}{\to }\left[\begin{array}{ccccc}{x}_{11}& x_{12}& \dots & x_{1n}& b_1\\ & & & & \\ ⋮& & & & \\ \lambda x_{i1}+xji& \lambda x_{i2}+x_{j2}& \dots & x_{in}& b_i\lambda b_j\\ & & & & \\ ⋮& & & & \\ x_{i1}& x_{i2}& \dots & x_{in}& b_i\\ & & & & \\ ⋮& & & & \\ x_{k1}& x_{k2}& \dots & x_{kn}& b_k\\ & & & & \end{array}\right]$$

$R_i\leftrightarrow R_j$ (Swapping rows)

$$\left[\begin{array}{ccccc}{x}_{11}& x_{12}& \dots & x_{1n}& b_1\\ & & & & \\ ⋮& & & & \\ x_{j1}& x_{j2}& \dots & x_{jn}& b_j\\ & & & & \\ ⋮& & & & \\ x_{i1}& x_{i2}& \dots & x_{in}& b_i\\ & & & & \\ ⋮& & & & \\ x_{k1}& x_{k2}& \dots & x_{kn}& b_k\\ & & & & \end{array}\right]\stackrel{R_i\leftrightarrow R_j}{\to }\left[\begin{array}{ccccc}{x}_{11}& x_{12}& \dots & x_{1n}& b_1\\ & & & & \\ ⋮& & & & \\ x_{i1}& x_{i2}& \dots & x_{in}& b_i\\ & & & & \\ ⋮& & & & \\ x_{j1}& x_{j2}& \dots & x_{jn}& b_j\\ & & & & \\ ⋮& & & & \\ x_{k1}& x_{k2}& \dots & x_{kn}& b_k\\ & & & & \end{array}\right]$$

Theorems

• Row Operations Preserve Solution Sets of Linear Systems Theorem
• Row Operations Example