Find the rank of the following matrix.

Question

$A= \left[ \begin{array}{ccc} 3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{array} \right]$

Applying, $R_{3}-\frac{1}{2}R_{2}$

~ $A= \left[ \begin{array}{ccc} 3 & -1 & 2 \\ -6 & 2 & 4 \\ 0 & 0 & 0 \end{array} \right]$

Applying, $R_{2}+2R_{1}$

~ $A= \left[ \begin{array}{ccc} 3 & -1 & 2 \\ 0 & 0 & 8 \\ 0 & 0 & 0 \end{array} \right]$

Answer 1

Row reduce the matrix and count the nonzero rows that remain.

Answer 2

Since you have a row of zeros, the rank is at most $2$: two non-zero rows. If the matrix can be reduced further, you may have another row of zeros, in which case one non-zero row remains: rank 1.

ADDED: Given your work/edit, how many NON-ZERO rows remain? That gives you the rank of the original matrix.

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For a "refresher", see this link, with various equivalent definitions of the rank of a matrix.