$$\text{Evaluate } \begin{vmatrix} 13 & 16 & 19 \\ 14 & 17 & 20 \\ 15 & 18 & 21 \end{vmatrix}$$

This determinant has a hidden structure: notice that each row is an arithmetic sequence, and each column is also an arithmetic sequence. This means the rows are linearly dependent—you can express one row as a combination of the others. When rows are dependent, the determinant collapses to zero. Rather than expanding directly, look for a row operation that reveals this dependency.

This determinant has a hidden structure: notice that each row is an arithmetic sequence, and each column is also an arithmetic sequence. This means the rows are linearly dependent—you can express one row as a combination of the others. When rows are dependent, the determinant collapses to zero. Rather than expanding directly, look for a row operation that reveals this dependency.