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"`