what is the determinant of a matrix?

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If someone could enlighten me,that would be nice. I am practicing for an upcoming math exam,and for the life of me I can not wrap my head around this as a concept. (I can calculate it, and after that do the fancy stuffs with it,but no scientific explanation helped me to understand what the hell a determinant is actually 😀 .

In: Mathematics

Its basically a Volume. If you have a 3×3 matrix it is the volume of the parallelepiped bounded by the origin and each of the vectors, with the other points being various sums of the vectors.

With a 2×2 its the area of a parallelogram where the points are the two vectors, the origin and the sum of the two vectors.

In 4+ dimensions it gets more abstract.

If you have an nxn transformation matrix, it will transform an nx1 vector. If you had am entire shape though and transformed each of its vertices to get another shape, the scale factor (e.g. of the areas in 2D, or volume in 3D), will be equal to the determinant of the transformation matrix.

It’s in a sense a bit like the size or power of a transformation matrix, at least how I think about it.

If the determinant is zero then it tells you that it maps all points to the origin. If the matrix represented a the coefficients of a set of simultaneous equations then a zero determinant would tell you there is no solution/no unique solution.