Solution: The determinant is 0 because the rows are linearly dependent.
Steps: 7
Step 1: Write down the matrix for which you want to calculate the determinant: \( A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{pmatrix} \).
Step 2: Identify the rows of the matrix: Row 1 is (1, 2, 1), Row 2 is (2, 3, 1), and Row 3 is (3, 4, 1).
Step 3: Check if the rows are linearly dependent. This means we need to see if one row can be formed by a combination of the others.
Step 4: Notice that Row 3 (3, 4, 1) can be formed by adding Row 1 (1, 2, 1) and Row 2 (2, 3, 1): (1 + 2, 2 + 3, 1 + 1) = (3, 4, 2).
Step 5: Since Row 3 can be expressed as a combination of Row 1 and Row 2, the rows are linearly dependent.
Step 6: When the rows of a matrix are linearly dependent, the determinant is 0.
Step 7: Therefore, the determinant of the matrix is 0.
