Chapter 6

Determinants6.1 Basic Techniques And Properties

6.1.1 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 6.1.1 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 6.1.2 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 6.1.3 Suppose A is a 3×3 matrix. The i jth minor, denoted as minor(A)i j , is thedeterminant of the 2×2 matrix which results from deleting the ith row and the jth column.

Example 6.1.4 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.

Definition 6.1.5 Suppose A is a 3×3 matrix. The i jth cofactor is defined to be (−1)i+ j×(i jth minor

). In words, you multiply (−1)i+ j times the i jth minor to get the i jth cofactor.

The cofactors of a matrix are so important that special notation is appropriate when re-ferring to them. The i jth cofactor of a matrix A will be denoted by cof(A)i j . It is also

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