Question:

Step 1: Understand the definition of a triangular matrix. A triangular matrix is a special type of square matrix where all the elements either above or below the main diagonal are zero. This means all non-zero entries are concentrated on or to one side of the main diagonal. Step 2: Relate the concept of rank to the structure of a triangular matrix. The rank of a matrix is the maximum number of linearly independent row vectors or column vectors. For a triangular matrix, the elements along the main diagonal are pivotal in determining its rank. If all diagonal elements are non-zero, the matrix is invertible, and its rank is full (equal to its dimension). If some diagonal elements are zero, the matrix is singular, and its rank is less than its dimension. Step 3: Determine the rank of a triangular matrix based on its diagonal elements. For any triangular matrix (upper or lower), the rank is equal to the number of non-zero elements on its main diagonal. This property holds because each non-zero diagonal element corresponds to a linearly independent row and column vector when considering the reduced row echelon form. If a diagonal element is zero, it potentially leads to a linearly dependent row or column. Step 4: Conclude the final answer. Therefore, for a triangular matrix, the rank is the number of non-zero diagonal elements. The final answer is $\boxed{\text{number of non-zero diagonal elements}}$.