The determinant is connected to many of the key ideas in linear algebra.

The determinant satisfies a number of useful properties, among them
(a)
(Determinants commute with products) If are two square matrices of the same dimensions, then .
(b)
(Test for singularity) If is a numerical square matrix, then is singular iff . Alternatively, is invertible as a matrix iff is invertible as a number.
(c)
(Determinant of triangular matrices) If is triangular (either upper or lower), then the product of the diagonal entries.
(d)
(Invariance under transposition) .
(e)
(Determinants of elementary matrices)
(i)
For any type I elementary matrix , .
(ii)
For any type II elementary matrix , .
(iii)
For any type III elementary matrix , .
(f)
(Cofactor expansions along rows or columns) If is an matrix, then
(i)
(Rows) For any , . Moreover, if , then .
(ii)
(Columns) For any , . Moreover, if , then .

These properties can be used to produce algorithms for computing the determinant of a numerical matrix that are considerably more efficient than direct application of the definition (either using cofactor expansion along the first row, or the combinatorial sum over all permutations).

We will say that two matrices of the same dimension are type III row equivalent if one can be derived from the other by using only type III row operations, and are type I-III row equivalent if one can be derived from the other by using only type I and type III row operations.

The following is not difficult to prove.

Prove this result.

A useful consequence of this theorem is the following algorithm for computing the determinant of a purely numerical matrix :

  • Using type I and III row operations, convert into an upper triangular matrix .
  • Compute as the product of its diagonal entries.
  • Count the number of type I operations you used in going from to . If it was an even number then . If it was an odd number then .

In practice the above procedure computes much more quickly than brute force application of either definition, and is used in many applications. However, it is only useful in the case is purely numerical (all of its entries are either real or complex numbers). For more general matrices that have non-numerical entries (for example, real-valued functions), Theorem 1 above no longer holds, and more sophisticated methods are needed for effectively computing the determinant.