Determine the rank of this matrix manually.

Question

I am trying to determine the rank of this matrix manually using the instructions found here: https://www.sangakoo.com/en/unit/rank-of-a-matrix-gaussian-method

The matrix:

[ [2, 0, -1, 0, 0, 0],
 [0, 6, -3, 0, 0, 0],
 [0, 0, 2, -3, 0, 0],
 [0, 0, 4, 0, -2, 0],
 [0, 0, 0, 3, 0, -1],
 [0, 0, 0, 0, 3, -3]
]

The rules mentioned in the article:

Only according to the definition, then, the cases in which a row can be discarded are:

All the elements are empty.
There are two equal rows.
A row is proportional to another.
A row is a linear combination of one or more.

Cases 1, 2 and 3 do not apply to any row in the matrix.

The answer is 5, implying that their is a row which fits case 4, however I can't come up with the linear combination.

Any help appreciated.

Answer 1

The first two rows have non-zero elements in columns where no other rows have non-zero elements, so they can’t be part of the solution. So you need to find a linear combination of three of the last four rows that equals the remaining one. Generate $0$s by forming appropriate combinations; e.g. start with the fourth row minus $2$ times the third to generate a $0$ in the third column, and continue like that.