INTRODUCTION

Suppose A is an n × n matrix. Associated with A is a number called the determinant of A and is denoted by det A. Symbolically, we distinguish a matrix A from the determinant of A by replacing the parentheses by vertical bars:

A determinant of an n × n matrix is said to be a determinant of order n. We begin by defining the determinants of 1 × 1, 2 × 2, and 3 × 3 matrices.

A Definition

For a 1 × 1 matrix A = (a), we have det A = |a| = a. For example, if A = (−5), then det A = |−5| = −5. In this case the vertical bars || around a number do not mean the absolute value of the number.

As a mnemonic, a determinant of order 2 is thought to be the product of the main diagonal entries of A minus the product of the other diagonal entries:

. (2)

For example, if A = , then det A = = 6(9) − (−3)(5) = 69.

The expression in (3) can be written in a more tractable form. By factoring, we have

det A = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) + a₁₂(−a₂₁a₃₃ + a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁).

But in view of (1), each term in parentheses is recognized as the determinant of a 2 × 2 matrix:

(4)

Observe that each determinant in (4) is a determinant of a submatrix of the matrix A and corresponds to its coefficient in the following manner: a₁₁ is the coefficient of the determinant of the submatrix obtained by deleting the first row and first column of A; a₁₂ is the coefficient of the negative of the determinant of the submatrix obtained by deleting the first row and second column of A; and finally, a₁₃ is the coefficient of the determinant of the submatrix obtained by deleting the first row and third column of A. In other words, the coefficients in (4) are simply the entries of the first row of A. We say that det A has been expanded by cofactors along the first row, with the cofactors of a₁₁, a₁₂, and a₁₃ being the determinants

Thus (4) is

det A = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃. (5)

In general, the cofactor of a_ij is the determinant

(6)

where M_ij is the determinant of the submatrix obtained by deleting the ith row and the jth column of A. The determinant M_ij is called a minor determinant. A cofactor is a signed minor determinant; that is, C_ij = M_ij when i + j is even and C_ij = −M_ij when i + j is odd.

A 3 × 3 matrix has nine cofactors:

Inspection of the above array shows that the sign factor +1 or −1 associated with a cofactor can be obtained from the checkerboard pattern:

(7)

Now observe that (3) can be rearranged and factored again as

(8)

which is the cofactor expansion of det A along the second column. It is left as an exercise to show from (3) that det A can also be expanded by cofactors along the third row:

det A = a₃₁C₃₁ + a₃₂C₃₂ + a₃₃C₃₃. (9)

We are, of course, suggesting in (5), (8), and (9) the following general result:

EXAMPLE 1 Cofactor Expansion Along the First Row

Evaluate the determinant of A = .

SOLUTION

Using cofactor expansion along the first row gives

det A = = 2C₁₁ + 4C₁₂ + 7C₁₃.

Now, the cofactors of the entries in the first row of A are

where the dashed lines indicate the row and column that are deleted. Thus,

≡

If a matrix has a row (or a column) containing many zero entries, then wisdom dictates that we evaluate the determinant of the matrix using cofactor expansion along that row (or column). Thus, in Example 1, had we expanded the determinant of A using cofactors along, say, the second row, then

EXAMPLE 2 Cofactor Expansion Along the Third Column

Evaluate the determinant of A = .

SOLUTION

Since there are two zeros in the third column, we expand by cofactors of that column:

Carrying the above ideas one step further, we can evaluate the determinant of a 4 × 4 matrix by multiplying the entries in a row (or column) by their corresponding cofactors and adding the products. In this case, each cofactor is a signed minor determinant of an appropriate 3 × 3 submatrix. The following theorem, which we shall give without proof, states that the determinant of any n × n matrix A can be evaluated by means of cofactors.

The sign factor pattern for the cofactors illustrated in (7) extends to matrices of order greater than 3:

EXAMPLE 3 Cofactor Expansion Along the Fourth Row

Evaluate the determinant of the matrix

SOLUTION

Since the matrix has two zero entries in its fourth row, we choose to expand det A by cofactors along that row:

det A = = (1)C₄₁ + 0C₄₂ + 0C₄₃ + (−4)C₄₄, (10)

where .

We then expand both these determinants by cofactors along the second row:

C₄₁ = (−1) = 18

Therefore (10) becomes

det A = = (1)(18) + (−4)(−4) = 34.

You should verify this result by expanding det A by cofactors along the second column. ≡

8.4 Exercises Answers to selected odd-numbered problems begin on page ANS-18.

In Problems 1–4, suppose

.

Evaluate the indicated minor determinant or cofactor.

1. M₁₂
2. M₃₂
3. C₁₃
4. C₂₂

In Problems 5–8, suppose

Evaluate the indicated minor determinant or cofactor.

5. M₃₃
6. M₄₁
7. C₃₄
8. C₂₃

In Problems 9–14, evaluate the determinant of the given matrix.

9. (−7)
10. (2)
11. 
12. 
13. 
14. 

In Problems 15–28, evaluate the determinant of the given matrix by cofactor expansion.

In Problems 29 and 30, find the values of λ that satisfy the given equation.