Determine the rank of the matrix. \[ \left[\begin{array}{rrrrr} 1 & 0 & -3 & 0 & 4 \\ 0 & 1 & -4 & 0 & 2 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \] 4 5 2 3

Step 1 ：Determine the rank of the matrix. The matrix is given as: \[ \left[\begin{array}{rrrrr} 1 & 0 & -3 & 0 & 4 \ 0 & 1 & -4 & 0 & 2 \ 0 & 0 & 0 & 1 & 1 \ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

Step 2 ：The rank of a matrix is the maximum number of linearly independent rows or columns. In this case, we can see that the last row is all zeros, which means it's not contributing to the rank.

Step 3 ：The other three rows are linearly independent (none of them can be written as a linear combination of the others), so the rank of the matrix is 3.

Step 4 ：Final Answer: The rank of the matrix is \(\boxed{3}\).