The determinant of a square real matrix is a real number that is a
special combination of the elements in the matrix. There is a general formula
for a determinant which uses a cofactor expansion. This formula is not
efficient for large matrices. If a large matrix is reduced to echelon form
(using only row replacements and interchanges),
then the determinant can be found by taking the product (with an appropriate
choice of sign) of the diagonal entries. This works because row replacement
operations do not change the determinant, and row interchanges only change the
sign of the determinant. Some important properties of the determinant:
transposing a matrix does not change its determinant;
the determinant of a product is the product of the determinants; the
determinant of the inverse of a matrix is the reciprocal of the determinant
of a matrix. A matrix is invertible if and only if the determinant is nonzero.
Although linear systems are usually solved without the use determinants,
the determinant formula for the solution to a linear system (Cramer's rule)
provides a compact formula that is useful in many applications. Cramer's
rule can also be used to find a formula for the inverse of a matrix. One
application of determinants is the computation of areas and volumes.
The most important parts of this lesson are the general formulas for the
determinant, some determinant properties, and Cramer's rule.