Find the rank of A = [[1, 2, 3], [2, 4, 6], [0, 1, 1]].

Step 1: Write the matrix.

A = [[1, 2, 3], [2, 4, 6], [0, 1, 1]]

The rank is the number of pivot rows after row reduction.

Step 2: Eliminate the first entry in row 2.

Use:
R2 → R2 - 2R1

The second row becomes:
[2, 4, 6] - 2[1, 2, 3] = [0, 0, 0]

So the matrix becomes:
[[1, 2, 3], [0, 0, 0], [0, 1, 1]]

Step 3: Move the nonzero row above the zero row.

Swap rows 2 and 3:

[[1, 2, 3], [0, 1, 1], [0, 0, 0]]

Step 4: Identify the pivot rows.

Row 1 contains a pivot in column 1.
Row 2 contains a pivot in column 2.
Row 3 is a zero row.

There are 2 pivot rows.

Step 5: State the rank.

Therefore:
rank(A) = 2.