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