Divisibility of a determinant

Task number: 2603

Numbers 697, 476 and 969 are divisible by 17. Prove that 17 divides the determinant of the following matrix \( \begin{pmatrix} 6 & 9 & 7\\ 4 & 7 & 6\\ 9 & 6 & 9\\ \end{pmatrix} \) without calculating it.

  • Resolution

    Follows from the linearity of the determinant. The numbers 697, 476 and 969 could be obtained, when the 100-multiple of the first column and 10-multiple of the second are added to the last one.

    Formally: \( \begin{vmatrix} 6 & 9 & 7\\ 4 & 7 & 6\\ 9 & 6 & 9\\ \end{vmatrix} = \begin{vmatrix} 6 & 9 & 697\\ 4 & 7 & 476\\ 9 & 6 & 969\\ \end{vmatrix} = 17\cdot \begin{vmatrix} 6 & 9 & \frac{697}{17}\\[1ex] 4 & 7 & \frac{476}{17}\\[1ex] 9 & 6 & \frac{969}{17}\\ \end{vmatrix} \)

    The last matrix contains only integers, and hence its determinant is also an integer.

Difficulty level: Easy task (using definitions and simple reasoning)
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