174 CHAPTER 10. EIGENVALUES AND EIGENVECTORS

10.2 An Introduction to DeterminantsHere in this section, I will summarize the main properties of determinants without detailedproofs. Proofs are presented later in the book. The idea is that you get used to using themfirst.

10.2.1 Cofactors and 2×2 DeterminantsLet A be an n× n matrix. The determinant of A, denoted as det(A) is a number. If thematrix is a 2×2 matrix, this number is very easy to find.

Definition 10.2.1 Let A =

(a bc d

). Then det(A) ≡ ad− cb. The determinant is also

often denoted by enclosing the matrix with two vertical lines. Thus

det

(a bc d

)=

∣∣∣∣∣ a bc d

∣∣∣∣∣ .Example 10.2.2 Find det

(2 4−1 6

).

From the definition this is just (2)(6)− (−1)(4) = 16.Having defined what is meant by the determinant of a 2×2 matrix, what about a 3×3

matrix?

Definition 10.2.3 Suppose A is a 3× 3 matrix. The i jth minor, denoted as minor(A)i j ,

is the determinant of the 2× 2 matrix which results from deleting the ith row and the jth

column.

Example 10.2.4 Consider the matrix 1 2 34 3 23 2 1

 .

The (1,2) minor is the determinant of the 2× 2 matrix which results when you delete thefirst row and the second column. This minor is therefore

det

(4 23 1

)=−2.

The (2,3) minor is the determinant of the 2× 2 matrix which results when you delete thesecond row and the third column. This minor is therefore

det

(1 23 2

)=−4.