Question

What is the determinant of a matrix?

Answer 1

See explanation...

Explanation:

An `n xx n` matrix represents a linear transformation (reflection, rotation, stretching, shearing, etc.) in `n` dimensional space.

The determinant is basically the factor by which a matrix scales the area/volume/hypervolume - with negative value if reflection is involved. If the result is zero then the matrix collapses `n` dimensions to fewer dimensions and the matrix is not invertible.

For example, in `2` dimensions, the determinant of a matrix `M` is the area of the quadrilateral obtained by transforming the unit square by multiplying its coordinates by `M`.

For `2xx2` matrices we can calculate the determinant as follows:

`abs((a,b),(c,d)) = ad - bc`

For `3xx3` matrices we can calculate the determinant as follows:

`abs((a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)) =a_11 abs((a_22,a_23),(a_32,a_33)) + a_12 abs((a_23,a_21),(a_33,a_31)) + a_13 abs((a_21,a_22),(a_31,a_32))`

A similar process can be used on larger matrices, but there are quicker methods.