What is the value of the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)?

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Q: What is the value of the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)?

Solution: The determinant is 0 because the first column is repeated.

Steps: 5

Step 1: Identify the matrix for which we need to find the determinant: \( A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \).
Step 2: Look at the first column of the matrix. It is \( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \).
Step 3: Notice that the first column is repeated in the sense that all entries are the same (1).
Step 4: Recall that if any two columns (or rows) of a matrix are identical or proportional, the determinant of that matrix is 0.
Step 5: Since the first column is repeated, we conclude that the determinant of the matrix is 0.