21.1.2 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 21.1.15 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 21.1.16 Let M be an upper (lower) triangular matrix. Then det
is obtained by
taking the product of the entries on the main diagonal.
Example 21.1.17 Let
From the above corollary, it suffices to take the product of the diagonal elements. Thus
Without using the corollary, you could expand along the first column.
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 equals
which is just the product of the entries down the main diagonal of the original matrix. It works this way in