For an n×n matrix A, the determinant of A denotes the magnification of the volume of n-dimensional parallelopiped.
More formally, the following are the determinants of 2×2 and 3×3 matrices.
If A=(a1,a2) is a 2×2 matrix, detA means the transformed area from the unit area. If A=(a1,a2,a3) is a 3×3 matrix, detA means the transformed volume form the unit volume.
For the arbitrary column vector ai of A, adding kai to another column vector aj does not change detA. For example, det(a1,a2,a3)=det(a1,a2,a3+ca2) for c∈R. Consider detA as a deck, then its volume is not changed although it is pushed to some directions.
The determinant of an upper triangular matrix A is the product of all the diagonal elements of A.
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