How to know the most efficient way of finding the rank of matrix and how do I apply it to this example

Question

For the matrix $ \begin{bmatrix} 1 & 2 & -3 \\ 1 & -2 & 3 \\ 4 & 8 & -12 \\ 1 & -1 & 5 \end{bmatrix} $

The reduced row echelon I obtained is $ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} $

However, the answer is that the rank of this matrix is 2. How do I know if the rank is indeed 2?

Answer 1

Doing Gaussian Elimination, you get $$ \begin{split} \begin{bmatrix} 1 & 2 & -3 \\ 1 & -2 & 3 \\ 4 & 8 & -12 \\ 1 & -1 & 5 \end{bmatrix} &\to \begin{bmatrix} 1 & 2 & -3 \\ 0 & 4 & -6 \\ 0 & 0 & 0 \\ 0 & -3 & 8 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & -3 \\ 0 & 1 & -3/2 \\ 0 & 0 & 0 \\ 0 & -3 & 8 \end{bmatrix} \\ &\to \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \end{split} $$ so you are right, and the rank is 3. Must be a typo in the book.