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I know that we may add any row to any other in a determinant and its value remains the same. This is clear enough since elementary matrices corresponding to row and column operations have determinant 1.

Is there an explanation of this fact in terms of geometric ideas such as volume and linear transformations?

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Think of the area of a rectangle (ABCD in the figure). Now, take one of the sides of the rectangle (DC) and shift it along the line it lies on so that the sides not parallel to the shifting one slant at an ever sharper angle (DC $\to$ FE). This process alters the shape of the figure into a parallelogram, but it does not alter the area of the figure.

Rectangle gets slanted into parallelogram

That's exactly the operation you do when you add one of the rows multiplied by a factor to another row of the determinant. The rows (or columns) correspond to the vectors $\vec{AD}$ and $\vec{AB}$. If you add $\vec{DF}$ to $\vec{AD}$, you obtain the new row $\vec{AF}$. Thus the determinant represents now the area of the parallelogram, which is the same as that of the rectangle.

This generalizes to higher dimensions.

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