Chapter 27

Determinants

27.1 Basic Techniques And PropertiesTo begin with, the basic computations and properties of determinants are discussed. Afterthis, complete proofs are given for those who are interested.

Actually, determinants were studied before the modern theory of linear algebra. Theyare very important in differential equations. Much of what was done earlier concerningeigenvalues and eigenvectors could have been presented without determinants, but thingslike the Wronskian are given to be certain determinants and so it is important to discussthem.

27.1.1 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 27.1.1 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 27.1.2 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 27.1.3 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.

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