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(B) Let $\Delta = \left| \begin{array}{ccc} 31 & 37 & 92 \ 31 & 58 & 71 \ 31 & 105 & 24 \end{array} ight|$.Applying the row operations $R_2 	o R_

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$0$

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(B) Let $\Delta = \left| \begin{array}{ccc} 31 & 37 & 92 \ 31 & 58 & 71 \ 31 & 105 & 24 \end{array} ight|$.Applying the row operations $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_1$:$\Delta = \left| \begin{array}{ccc} 31 & 37 & 92 \ 31-31 & 58-37 & 71-92 \ 31-31 & 105-37 & 24-92 \end{array} ight|$$\Delta = \left| \begin{array}{ccc} 31 & 37 & 92 \ 0 & 21 & -21 \ 0 & 68 & -68 \end{array} ight|$Since the second and third rows are proportional (or by factoring out $21$ from $R_2$ and $68$ from $R_3$,we get identical columns),the determinant is $0$.Specifically,$R_3 = \frac{68}{21} R_2$,so the rows are linearly dependent.Therefore,$\Delta = 0$.

The value of the determinant $\left| \begin{array}{ccc} 31 & 37 & 92 \\ 31 & 58 & 71 \\ 31 & 105 & 24 \end{array} \right|$ is

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