What exactly is the determinant of a matrix and why is it important? (Yeah sure I know how to calculate a 2x2 matrix's determinant and use it for the inverse - BUT WHY??)

I'd also like to know (after defining it though) how you can calculate the determinant of a 3x3 matrix (or any matrix - I guess determinants are only for square matrices?)

Thanks to anyone who replies

Determinants are very important for working with matrices. Firstly, for any square matrix M, det M = 0 < -> M is singular (i.e. has no inverse). Secondly, they can be used to compute the inverse of any non-singular square matrix M, since M^-1 = (adj M)/(det M) (you can see why det M =/= 0). Thirdly, the ability to calculate determinants is essential for finding eigenvalues and eigenvectors - these allow us to derive explicit formulae for powers of matrices (amongst other things).

There are definitions here or at Mathworld .

For a 3x3 matrix,

ê ê ê ê ê a11 a12 a13 a21 a22 a23 a31 a32 a33 ê ê ê ê ê = a11 ê ê ê a22 a23 a32 a33 ê ê ê -a12 ê ê ê a21 a23 a31 a33 ê ê ê +a13 ê ê ê a21 a22 a31 a32 ê ê ê

or it can be expanded by any other row or column, e.g. by a₁₂ , a₂₂ and a₃₂ .

Paul

Thanks that really helped

If you have a 2x2 transformation matrix, then given an object with non-zero area - the determinant of the transformation represents the area scale factor of the object (from before transformation to after). Clearly, if a transformation matrix has a determinant of zero magnitude, then we are destroying area. We are infact reducing the coordinate system to a lower dimensions, destroying data - this is reasoning why a matrix is singular if it has a zero determinant; we've lost data we can never retrieve.

Julian

The determinant of a linear transformation (matrix) T is the unique number D(T) such that for (almost) all sets E of area m(E)

m(T(E))=D(T)m(E)

Thanks... I was aware of the fact Julian mentioned but thanks for explaining Julian and William