What is the determinant of a matrix?

1 Answer
Mar 10, 2016

Answer:

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.