Determinants of n x n Matrices
Leibniz formula
Given an n x n matrix , we can generalize the definition of the signed volume of parallelepiped to the "n-dimensional signed volume" of the so-called parallelotope formed by the column vectors in and use it as the definition of . Similar to what we did in the previous section, we write each column vector as a linear combination of the standard basis in and then expand by linearity. Without getting into detail computation, which is quite involved, we write down the following result:
where the sum runs through all the permutation of the indices and , the signature of , is either or , depending whether the permutation can be obtained by successively interchanging two indices an even or odd number of times. For example, when and , because can be obtained by interchanging 1 and 2, then 1 and 3 and finally 1 and 4 () i.e. interchanging odd number of times.
This is the Leibniz formula for the determinant of an n x n matrix.
However, this formula is seldom used for actual computation because it is too complicated to use. Fortunately, we have some other ways to compute determinants.
Recursive Formula for Determinants
We can express the determinant of an n x n matrix in terms of the determinants of submatrices of (matrices obtained by deleting one or several rows and/or columns of ). Let be the n-1 x n-1 submatrix of formed by deleting the ith row and jth column of . Then we have the following formula:
The term is called the -cofactor of , denoted by . Hence, for ,
is called a cofactor expansion across the ith row of A.
In fact, we can also compute the determinant by a cofactor expansion across the jth column of A:
The proofs of these formulas are lengthy and are thus omitted.
Example: Evaluate where .
We can choose the first column to do the cofactor expansion because most of its entries are zeros. We have
Then we choose the first column of the 4 x 4 matrix to do the cofactor expansion. We have
Then we choose the third column of the 3 x 3 matrix to do the factor expansion. We have
Exercise
Let . Find . This is an example of upper triangular matrices. What can you say about the determinant of an upper triangular matrix?