Evaluate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \)

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Q: Evaluate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \)

Solution: The determinant is 0 because the rows are linearly dependent.

Steps: 7

Step 1: Write down the matrix for which we want to find 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 2 can be obtained by adding Row 1 and Row 1 (1 + 1 = 2, 2 + 1 = 3, 1 + 0 = 1).
Step 5: Since Row 2 can be formed from Row 1, 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.