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7Matrices andDeterminants

7.4

Copyright © Cengage Learning. All rights reserved.

THE DETERMINANT OF A SQUARE MATRIX

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• Find the determinants of 2 2 matrices.

• Find minors and cofactors of square matrices.

• Find the determinants of square matrices.

What You Should Learn

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The Determinant of a 2 2 Matrix

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The Determinant of a 2 2 Matrix

Every square matrix can be associated with a real number called its determinant.

Determinants have many uses, and several will be discussed in this and the next section.

Historically, the use of determinants arose from special number patterns that occur when systems of linear equations are solved.

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The Determinant of a 2 2 Matrix

For instance, the system

a1x + b1y = c1

a2x + b2y = c2

has a solution

and

provided that a1b2 – a2b1 0. Note that the denominators of the two fractions are the same.

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The Determinant of a 2 2 Matrix

This denominator is called the determinant of the coefficient matrix of the system.

Coefficient Matrix Determinant

det(A) = a1b2 – a2b1

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The Determinant of a 2 2 Matrix

The determinant of the matrix A can also be denoted by vertical bars on both sides of the matrix, as indicated in the following definition.

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The Determinant of a 2 2 Matrix

In this text, det(A) and | A | are used interchangeably to represent the determinant of A.

Although vertical bars are also used to denote the absolute value of a real number, the context will show which use is intended.

A convenient method for remembering the formula for the determinant of a 2 2 matrix is shown in the following diagram.

Note that the determinant is the difference of the products of the two diagonals of the matrix.

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Example 1 – The Determinant of a 2 2 Matrix

Find the determinant of each matrix.

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Example 1 – Solution

= 2(2) – 1(–3)

= 4 + 3

= 7

= 2(2) – 4(1)

= 4 – 4

= 0

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Example 1 – Solution

= 0(4) – 2

= 0 – 3

= –3

cont’d

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The Determinant of a 2 2 Matrix

Notice in Example 1 that the determinant of a matrix can be positive, zero, or negative.

The determinant of a matrix of order 1 1 is defined simply as the entry of the matrix. For instance, if A = [–2], then det(A) = –2.

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Minors and Cofactors

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Minors and Cofactors

To define the determinant of a square matrix of order 3 3 or higher, it is convenient to introduce the concepts of minors and cofactors.

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Minors and Cofactors

In the sign pattern for cofactors shown below, notice that odd positions (where i + j is odd) have negative signs and even positions (where i + j is even) have positive signs.

Sign Pattern for Cofactors

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Example 2 – Finding the Minors and Cofactors of a Matrix

Find all the minors and cofactors of

Solution:

To find the minor M11, delete the first row and first column of A and evaluate the determinant of the resulting matrix.

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Example 2 – Solution

= –1(1) – 0(2)

= –1

Similarly, to find M12, delete the first row and second column.

= 3(1) – 4(2)

= –5

cont’d

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Example 2 – Solution

Continuing this pattern, you obtain the minors.

M11 = –1 M12 = –5 M13 = 4

M21 = 2 M22 = –4 M23 = –8

M31 = 5 M32 = –3 M33 = –6

Now, to find the cofactors, combine these minors with the checkerboard pattern of signs for a 3 3 matrix shown.

C11 = –1 C12 = 5 C13 = 4

C21 = –2 C22 = –4 C23 = 8

C31 = 5 C32 = 3 C33 = –6

cont’d

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The Determinant of a Square Matrix

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The Determinant of a Square Matrix

The definition below is called inductive because it uses

determinants of matrices of order n – 1 to define

determinants of matrices of order n.

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The Determinant of a Square Matrix

Try checking that for a 2 2 matrix

this definition of the determinant yields | A | = a1b2 – a2b1,

as previously defined.

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Example 3 – The Determinant of a Matrix of Order 3 3

Find the determinant of

Solution:

Note that this is the same matrix that was in Example 2. There you found the cofactors of the entries in the first row to be

C11 = –1, C12 = 5, and C13 = 4.

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Example 3 – Solution

So, by the definition of a determinant, you have

| A | = a11C11 + a12C12 + a13C13

= 0(–1) + 2(5) + 1(4)

= 14.

First-row expansion

cont’d