Augmented and Coefficient rank?

Question

I am confused by these 2 terms. Augmented rank and coefficient rank are they refer to the same thing or different? I know that that to find the rank of a matrix I have to reduce it to Echelon form first and then count the non-zero row.

Answer 1

A coefficient matrix only contains the coefficients of the variables of the linear equations. "Augmented" refers to the addition of a column (usually separated by a vertical line) of the constant terms of the linear equations.

For instance, given the system

\begin{align*} x+2y&=-3\\ -3x+y&=1 \end{align*}

the coefficient matrix is

$$\begin{bmatrix} 1 & 2 \\ -3 & 1 \end{bmatrix}$$

while the augmented matrix is

$$\left[\begin{array}{cc|c} 1 & 2 & -3\\ -3 & 1 & 1 \end{array}\right].$$