Visualising row addition on a 3×3 matrix
This applet shows how the determinant of a 3×3 matrix is unaffected by the elementary row operation of addition of a scalar multiple of a row to another row.
The determinant of a 3×3 matrix A is equal to the signed volume of the parallelepiped whose edges are given by the rows of A.
The parallelepiped in blue has edges given by the rows of the first matrix, and the parallelepiped in red has edges given by the rows of the second matrix, obtained by adding a multiple of row 1 to row 2. Use the sliders to change the multiples.
Both parallelepipeds share a base, and the height perpendicular to the base is always the same, regardless of the value of the scalar. This shows that the two parallelepipeds have the same volume.