412 CHAPTER 19. EIGENVALUES AND EIGENVECTORS

You should regard the above claim that you always get the same answer by pickingany row or column with considerable skepticism. It is incredible and not at all obvious.However, it requires a little effort to establish it. This is done in the section on the theory ofthe determinant, Section 20 which is presented later. This is summarized in the followingtheorem whose conclusion is incredible.

Theorem 19.2.10 Expanding the n× n matrix along any row or column alwaysgives the same answer so the above definition is a good definition.

Example 19.2.11 Expand

∣∣∣∣∣∣∣∣1 2 −1 12 3 1 11 1 0 01 2 3 1

∣∣∣∣∣∣∣∣ along first column.

It is

1

∣∣∣∣∣∣3 1 11 0 02 3 1

∣∣∣∣∣∣−2

∣∣∣∣∣∣2 −1 11 0 02 3 1

∣∣∣∣∣∣+1

∣∣∣∣∣∣2 −1 13 1 12 3 1

∣∣∣∣∣∣−1

∣∣∣∣∣∣2 −1 13 1 11 0 0

∣∣∣∣∣∣= 0

19.2.2 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 19.2.12 A matrix M, is upper triangular if Mi j = 0 whenever i > j.Thus such a matrix equals zero below the main diagonal, the entries of the form Mii, asshown. 

∗ ∗ · · · ∗

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 19.2.13 Let M be an upper (lower) triangular matrix. Then det(M) is ob-tained by taking the product of the entries on the main diagonal.

Example 19.2.14 Let

A =

1 2 3 770 2 6 70 0 3 33.70 0 0 −1

Find det(A) .