19.2.2 The Determinant Of A Triangular Matrix
Notwithstanding the difficulties involved in using the method of Laplace expansion,
certain types of matrices are very easy to deal with.
Definition 19.2.15 A matrix M, is upper triangular if Mij = 0 whenever
i > j. Thus such a matrix equals zero below the main diagonal, the entries of the form
Mii, as shown.
A lower triangular matrix is defined similarly as a matrix for which all entries above the
main diagonal are equal to zero.
You should verify the following using the above theorem on Laplace expansion.
Corollary 19.2.16 Let M be an upper (lower) triangular matrix. Then
is obtained by taking the product of the entries on the main diagonal.
Example 19.2.17 Let
From the above corollary, it suffices to take the product of the diagonal elements.
Without using the corollary, you could expand
along the first column. This gives
and the only nonzero term in the expansion is
Now expand this along the first column to obtain
Next expand this last determinant along the first column to obtain the above
which is just the product of the entries down the main diagonal of the original matrix. It
works this way in general.