8.5 Properties of Determinants

INTRODUCTION

In this section we are going to consider some of the many properties of determinants. Our goal in the discussion is to use these properties to develop a means of evaluating a determinant that is an alternative to cofactor expansion.

Properties

The first property states that the determinant of an n × n matrix and its transpose are the same.

THEOREM 8.5.1 Determinant of a Transpose

If A^T is the transpose of the n × n matrix A, then det A^T = det A.

For example, for the matrix A = , we have A^T = . Observe that

det A = = −41 and det A^T = = −41.

Since transposing a matrix interchanges its rows and columns, the significance of Theorem 8.5.1 is that statements concerning determinants and the rows of a matrix also hold when the word row is replaced by the word column.

THEOREM 8.5.2 Two Identical Rows

If any two rows (columns) of an n × n matrix A are the same, then det A = 0.

EXAMPLE 1 Matrix with Two Identical Rows

Since the second and third columns in the matrix A = are the same, it follows from Theorem 8.5.2 that

det A = = 0.

You should verify this by expanding the determinant by cofactors. ≡

THEOREM 8.5.3 Zero Row or Column

If all the entries in a row (column) of an n × n matrix A are zero, then det A = 0.

PROOF:

Suppose the ith row of A consists of all zeros. Hence all the products in the expansion of det A by cofactors along the ith row are zero and consequently det A = 0. ≡

For example, it follows immediately from Theorem 8.5.3 that

THEOREM 8.5.4 Interchanging Rows

If B is the matrix obtained by interchanging any two rows (columns) of an n × n matrix A, then det B = −det A.

For example, if B is the matrix obtained by interchanging the first and third rows of A = , then from Theorem 8.5.4 we have

det B = = −det A.

You should verify this by computing both determinants.

THEOREM 8.5.5 Constant Multiple of a Row

If B is the matrix obtained from an n × n matrix A by multiplying a row (column) by a nonzero real number k, then det B = k det A.

PROOF:

Suppose the entries in the ith row of A are multiplied by the number k. Call the resulting matrix B. Expanding det B by cofactors along the ith row then gives

≡

EXAMPLE 2 Theorems 8.5.5 and 8.5.2

(a)

(b) ≡

THEOREM 8.5.6 Determinant of a Matrix Product

If A and B are both n × n matrices, then det AB = det A · det B.

In other words, the determinant of a product of two n × n matrices is the same as the product of the determinants.

EXAMPLE 3 Determinant of a Matrix Product

Suppose A = and B = . Then AB = . Now det AB = −24, det A = −8, det B = 3, and so we see that

det A · det B = (−8)(3) = −24 = det AB. ≡

THEOREM 8.5.7 Determinant Is Unchanged

Suppose B is the matrix obtained from an n × n matrix A by multiplying the entries in a row (column) by a nonzero real number k and adding the result to the corresponding entries in another row (column). Then det B = det A.

EXAMPLE 4 A Multiple of a Row Added to Another

Suppose A = and suppose the matrix B is defined as that matrix obtained from A by the elementary row operation

Expanding by cofactors along, say, the second column, we find det A = 45 and det B = 45. You should verify this result. ≡

THEOREM 8.5.8 Determinant of a Triangular Matrix

Suppose A is an n × n triangular matrix (upper or lower). Then

det A = a₁₁a₂₂ . . . a_nn,

where a₁₁, a₂₂, . . ., a_nn are the entries on the main diagonal of A.

PROOF:

We prove the result for a 3 × 3 lower triangular matrix

Expanding det A by cofactors along the first row gives

det A = a₁₁ = a₁₁(a₂₂a₃₃ − 0 · a₃₂) = a₁₁a₂₂a₃₃. ≡

EXAMPLE 5 Determinant of a Triangular Matrix

(a) The determinant of the lower triangular matrix

(b) The determinant of the diagonal matrix A = is

det A = = (−3) · 6 · 4 = −72. ≡

Row Reduction

Evaluating the determinant of an n × n matrix by the method of cofactor expansion requires a Herculean effort when the order of the matrix is large. To expand the determinant of, say, a 5 × 5 matrix with nonzero entries requires evaluating five cofactors that are determinants of 4 × 4 submatrices; each of these in turn requires four additional cofactors that are determinants of 3 × 3 submatrices, and so on. There is a more practical (and programmable) method for evaluating the determinant of a matrix. This method is based on reducing the matrix to a triangular form by row operations and the fact that determinants of triangular matrices are easy to evaluate (see Theorem 8.5.8).

EXAMPLE 6 Reducing a Determinant to Triangular Form

Evaluate the determinant of A = .

SOLUTION

Our final theorem concerns cofactors. We saw in Section 8.4 that a determinant det A of an n × n matrix A could be evaluated by cofactor expansion along any row (column). This means that the n entries a_ij of a row (column) are multiplied by the corresponding cofactors C_ij and the n products are added. If, however, the entries a_ij of a row (a_ij of a column) of A are multiplied by the corresponding cofactors C_kj of a different row (C_ik of a different column), the sum of the n products is zero.

THEOREM 8.5.9 A Property of Cofactors

Suppose A is an n × n matrix. If a_i1, a_i2, . . . , a_in are the entries in the ith row and C_k1, C_k2, . . . , C_kn are the cofactors of the entries in the kth row, then

a_i1C_k1 + a_i2C_k2 + . . . + a_inC_kn = 0 for i ≠ k.

If a_1j, a_2j, . . . , a_nj are the entries in the jth column and C_1k, C_2k, . . ., C_nk are the cofactors of the entries in the kth column, then

a_1jC_1k + a_2jC_2k + . . . + a_njC_nk = 0 for j ≠ k.

PROOF:

We shall prove the result for rows. Let B be the matrix obtained from A by letting the entries in the ith row of A be the same as the entries in the kth row—that is, a_i1 = a_k1, a_i2 = a_k2, . . ., a_in = a_kn. Since B has two rows that are the same, it follows from Theorem 8.5.2 that det B = 0. Cofactor expansion along the kth row then gives the desired result:

≡

EXAMPLE 7 Cofactors of Third Row/Entries of First Row

Consider the matrix A = . Suppose we multiply the entries of the first row by the cofactors of the third row and add the results; that is,

≡

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

In Problems 1–10, state the appropriate theorem(s) in this section that justifies the given equality. Do not expand the determinants by cofactors.

In Problems 11−16, evaluate the determinant of the given matrix using the result

In Problems 17–20, evaluate the determinant of the given matrix without expanding by cofactors.

In Problems 21 and 22, verify that det A = det A^T for the given matrix A.

Consider the matrices
A = and B = .

Verify that det AB = det Adet B.
Suppose A is an n × n matrix such that A² = I. Then show that det A = ±1.
Suppose A is an n × n matrix such that A² = A. Then show that either det A = 0 or det A = 1.
If A and B are n × n matrices, then prove or disprove that det AB = det BA.
Consider the matrix
A = .

Without expanding, evaluate det A.
Consider the matrix
A = .

Without expanding, show that det A = 0.

In Problems 29–36, use the procedure illustrated in Example 6 to evaluate the determinant of the given matrix.

By proceeding as in Example 6, show that
= (b − a)(c − a)(c − b).
Evaluate . [Hint: See Problem 37.]

In Problems 39 and 40, verify Theorem 8.5.9 by evaluating a₂₁C₁₁ + a₂₂C₁₂ + a₂₃C₁₃ and a₁₃C₁₂ + a₂₃C₂₂ + a₃₃C₃₂ for the given matrix.

Let A = and B = . Verify that
det(A + B) ≠ det A + det B.
Suppose A is a 5 × 5 matrix for which det A = −7. What is the value of det(2A)?
An n × n matrix A is said to be skew-symmetric if A^T = −A. If A is a 5 × 5 skew-symmetric matrix, show that det A = 0.
It takes about n! multiplications to evaluate the determinant of an n × n matrix using expansion by cofactors, whereas it takes about n³/3 arithmetic operations using the row-reduction method. Compare the number of operations for both methods using a 25 × 25 matrix.