27.1. BASIC TECHNIQUES AND PROPERTIES 501

Definition 27.1.13 Let A be an n× n matrix where n ≥ 2 and suppose the determinant ofan (n−1)× (n−1) has been defined. Then

det(A) =n

∑j=1

ai j cof(A)i j =n

∑i=1

ai j cof(A)i j . (27.1)

The first formula consists of expanding the determinant along the ith row and the secondexpands the determinant along the jth column.

Theorem 27.1.14 Expanding the n× n matrix along any row or column always gives thesame answer so the above definition is a good definition.

27.1.2 The Determinant Of A Triangular MatrixNotwithstanding the difficulties involved in using the method of Laplace expansion, certaintypes of matrices are very easy to deal with.

Definition 27.1.15 A matrix M, is upper triangular if Mi j = 0 whenever i > j. Thus sucha matrix equals zero below the main diagonal, the entries of the form Mii, as shown.

∗ ∗ · · · ∗

0 ∗. . .

......

. . .. . . ∗

0 · · · 0 ∗

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 27.1.16 Let M be an upper (lower) triangular matrix. Then det(M) is obtainedby taking the product of the entries on the main diagonal.

Example 27.1.17 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

1

∣∣∣∣∣∣∣2 6 70 3 33.70 0 −1

∣∣∣∣∣∣∣+0(−1)2+1

∣∣∣∣∣∣∣2 3 770 3 33.70 0 −1

∣∣∣∣∣∣∣+0(−1)3+1

∣∣∣∣∣∣∣2 3 772 6 70 0 −1

∣∣∣∣∣∣∣+0(−1)4+1

∣∣∣∣∣∣∣2 3 772 6 70 3 33.7

∣∣∣∣∣∣∣