I'm learning Linear Algebra right now and at current about how to find rank of any given matrix. I'm confused in a problem from sometime. I don't know where I am wrong with the solution. Please have a look at the images.
My Solution 
Solution given in my book
The book tells that the given matrix will be of 2 rank and I think my solution will show rank 1. Please help. Thankyou in advance.
Hint-$1$: Observe that $C_1 \& C_2$ are linearly independent. Reason there is no such $\alpha$ for which $C_2 = \alpha C_1$ ($1=\alpha\cdot0$). $$C_3 = -3C_2+C_1 \dots (1)$$ $$C_4 = C_1-C_2 \dots (2)$$
Hint-$2$: Use Elementary Row Operators' to get the Row Reduced Echelon form. Take any standard text book, and you will find it there.