6.1. BASIC TECHNIQUES AND PROPERTIES 111

A lower triangular matrix is defined similarly as a matrix for which all entries above themain diagonal are equal to zero.

You should verify the following using the above theorem on Laplace expansion.

Corollary 6.1.16 Let M be an upper (lower) triangular matrix. Then det(M) is obtainedby taking the product of the entries on the main diagonal.

Example 6.1.17 Let

A =

1 2 3 770 2 6 70 0 3 33.70 0 0 −1

Find det(A) .

From the above corollary, it suffices to take the product of the diagonal elements. Thusdet(A) = 1×2×3× (−1) =−6. Without using the corollary, you could expand along thefirst column. This gives

1det

 2 6 70 3 33.70 0 −1

+0(−1)2+1 det

 2 3 770 3 33.70 0 −1

+0(−1)3+1 det

 2 3 772 6 70 0 −1

+0(−1)4+1 det

 2 3 772 6 70 3 33.7

and the only nonzero term in the expansion is

1det

 2 6 70 3 33.70 0 −1

 .

Now expand this along the first column to obtain2×det

(3 33.70 −1

)+0(−1)2+1 det

(6 70 −1

)

+0(−1)3+1 det

(6 73 33.7

)= 1×2×det

(3 33.70 −1

)

Next expand this last determinant along the first column to obtain the above equals

1×2×3× (−1) =−6

which is just the product of the entries down the main diagonal of the original matrix. Itworks this way in general.