What is meant by the determinant of a matrix?

Assuming that we have a square matrix, then the determinant of the matrix is the determinant with the same elements.

Eg if we have a #2xx2# matrix:

# bb(A) = ( (a,b), (c,d) ) #

The the associated determinant given by

# D = | bb(A) | = | (a,b), (c,d) | = ad-bc#

To extend on Steve's explanation, the determinant of a matrix tells you whether or not the matrix is invertible. If the determinant is 0, the matrix is not invertible.

For example, let #A=((1,3),(-2,1))#. Then #det(A)=1(1)-3(-2)=7# so we know that #A^-1# exists.
If we let #B=((1,2),(-2,-4))#, #det(B)=1(-4)-2(-2)=0# so we know that #B^-1# doesn't exist.
Additionally, the determinant is involved in computing the inverse of a matrix. Given a matrix #A=((a,b),(c,d))#, #A^-1=1/det(A)((d,-b),(-c,a))#. From this, you can see why #A^-1# doesn't exist when #det(A)=0#.

The determinant is also used as a area/volume scale factor,

If we have a #2xx2 # matrix, #M#

Then if a particular shape of area #A# undergoes the transformation defined by the matrix #M# then the area of the new shape will be #det(M) A # or #|M|A #

Also

#det(M) = 0 <=> " M defined as being 'singular' , no inverse"#