Find the rank of the given matrix.

To find the rank of the matrix, we analyze the linear independence of its rows.

Step-by-Step Solution
The given matrix is:
A = | 1  5  6 |
| 2  3  4 |
|-1  2  2 |

Step 1: Perform Row Operations to reach Row Echelon Form
We keep Row 1 (R1) as the pivot:
R1: [1, 5, 6]

Perform R2 -> R2 - 2R1:
R2 = [2, 3, 4] - 2[1, 5, 6] = [2-2, 3-10, 4-12] = [0, -7, -8]

Perform R3 -> R3 + R1:
R3 = [-1, 2, 2] + [1, 5, 6] = [-1+1, 2+5, 2+6] = [0, 7, 8]

The matrix now looks like:
| 1   5   6 |
| 0  -7  -8 |
| 0   7   8 |

Step 2: Simplify further
Perform R3 -> R3 + R2:
R3 = [0, 7, 8] + [0, -7, -8] = [0, 0, 0]

The matrix in row echelon form is:
| 1   5   6 |
| 0  -7  -8 |
| 0   0   0 |

Step 3: Count non-zero rows
There are exactly 2 non-zero rows remaining. Therefore, the rank of the matrix is 2.