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212 CHAPTER 3 Determinants
49. If A and S are n × n matrices with S invertible, show that det(S−^1 AS) = det(A). [ Hint: Since S−^1 S = In, how are det(S−^1 ) and det(S) related?] 50. If det(A^3 ) = 0, is it possible for A to be invertible? Justify your answer. 51. Let E be an elementary matrix. Verify the formula for det(E) given in the text at the beginning of the proof of P8. 52. Show that x y 1 x 1 y 1 1 x 2 y 2 1
represents the equation of the straight line through the distinct points (x 1 , y 1 ) and (x 2 , y 2 ).
53. Without expanding the determinant, show that
1 x x^2 1 y y^2 1 z z^2
= (y − z)(z − x)(x − y).
54. If A is an n × n skew-symmetric matrix and n is odd, prove that det(A) = 0. 55. Let A = [ a 1 , a 2 ,... , a n] be an n × n matrix, and let b = c 1 a 1 + c 2 a 2 + · · · + cn a n, where c 1 , c 2 ,... , c (^) n are constants. If Bk denotes the matrix obtained from A by replacing the kth column vector by b , prove that
det(Bk ) = ck det(A), k = 1 , 2 ,... , n.
56. Let A be the general 4 × 4 matrix.
(a) Verify property P1 of determinants in the case when the first two rows of A are permuted.
(b) Verify property P2 of determinants in the case when row 1 of A is divided by k. (c) Verify property P3 of determinants in the case when k times row 2 is added to row 1.
57. For a randomly generated 5 × 5 matrix, verify that det(AT^ ) = det(A). 58. Determine all values of a for which 1 2 3 4 a 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 a 4 3 2 1
is invertible.
59. If
A =
determine all values of the constant k for which the linear system (A − kI 3 ) x = 0 has an infinite number of solutions, and find the corresponding solutions.
60. Use the determinant to show that
A =
is invertible, and use A−^1 to solve A x = b if b = [ 3 , 7 , 1 , − 4 ]T^.
3.3 Cofactor Expansions
We now obtain an alternative method for evaluating determinants. The basic idea is that we can reduce a determinant of order n to a sum of determinants of order n−1. Continuing in this manner, it is possible to express any determinant as a sum of determinants of order 2. This method is the one most frequently used to evaluate a determinant by hand, although the procedure introduced in the previous section whereby we use elementary row operations to reduce the matrix to upper triangular form involves less work in general. When A is invertible, the technique we derive leads to formulas for both A−^1 and the unique solution to A x = b. We first require two preliminary definitions.
DEFINITION 3.3.
Let A be an n × n matrix. The minor , M (^) ij , of the element aij , is the determinant of the matrix obtained by deleting the ith row vector and j th column vector of A.
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3.3 Cofactor Expansions 213
Remark Notice that if A is an n × n matrix, then Mij is a determinant of order n − 1. By convention, if n = 1, we define the “empty” determinant M 11 to be 1.
Example 3.3.2 If
A =
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
then, for example,
M 23 =
a 11 a 12 a 31 a 32 and M 31 = a 12 a 13 a 22 a 23
Example 3.3.3 Determine the minors M 11 , M 23 , and M 31 for
A =
Solution: Using Definition 3.3.1, we have
M 11 =
= 22 , M 23 =
= − 1 , M 31 =
DEFINITION 3.3.
Let A be an n × n matrix. The cofactor , Cij , of the element aij , is defined by
Cij = (− 1 ) i+j^ Mij ,
where Mij is the minor of aij.
From Definition 3.3.4, we see that the cofactor of aij and the minor of aij are the same if i + j is even, and they differ by a minus sign if i + j is odd. The appropriate sign in the cofactor Cij is easy to remember, since it alternates in the following manner:
- − + − + · · · − + − + − · · ·
- − + − + · · · .. .
Example 3.3.5 Determine the cofactors C 11 , C 23 , and C 31 for the matrix in Example 3.3.3.
Solution: We have already obtained the minors M 11 , M 23 , and M 31 in Example 3.3.3, so it follows that
C 11 = +M 11 = 22 , C 23 = −M 23 = 1 , C 31 = +M 31 = − 14.
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3.3 Cofactor Expansions 215
Solution:
(a) We have
2 3 4 1 −1 1 6 3 0
(b) We have
2 3 4 1 −1 1 6 3 0
Notice that (b) was easier than (a) in the previous example, because of the zero in column 3. Whenever one uses the cofactor expansion method to evaluate a determinant, it is usually best to select a row or column containing as many zeros as possible in order to minimize the amount of computation required.
Example 3.3.9 Evaluate 0 3 −1 0 5 0 8 2 7 2 5 4 6 1 7 0
Solution: In this case, it is easiest to use either row 1 or column 4. Choosing row 1, we have
0 3 −1 0 5 0 8 2 7 2 5 4 6 1 7 0
= −3 [2 ( 49 − 30 ) − 4 ( 35 − 48 ) + 0] − [5 ( 0 − 4 ) − 0 + 2 ( 7 − 12 )]
In evaluating the determinants of order 3 on the right side of the first equality, we have used cofactor expansion along column 3 and row 1, respectively. For additional practice, the reader may wish to verify our result here by cofactor expansion along a different row or column. Now we turn to the
Proof of the Cofactor Expansion Theorem: It follows from the definition of the determinant that det(A) can be written in the form
det(A) = ai 1 Cˆi 1 + a 12 Cˆi 2 + · · · + ain Cˆin (3.3.1)
where the coefficients Cˆij contain no elements from row i or column j. We must show that Cˆij = Cij where Cij is the cofactor of aij. Consider first a 11. From Definition 3.1.8, the terms of det(A) that contain a 11 are given by a 11
σ ( 1 , p 2 , p 3 ,... , p (^) n )a 2 p 2 a 3 p 3 · · · anp (^) n ,
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216 CHAPTER 3 Determinants
where the summation is over the (n − 1 )! distinct permutations of 2, 3 ,... , n. Thus,
C^ ˆ 11 =
σ ( 1 , p 2 , p 3 ,... , p (^) n )a 2 p 2 a 3 p 3 · · · anp (^) n.
However, this summation is just the minor M 11 , and since C 11 = M 11 , we have shown the coefficient of a 11 in det(A) is indeed the cofactor C 11. Now consider the element aij. By successively interchanging adjacent rows and columns of A, we can move aij into the (1, 1) position without altering the relative positions of the other rows and columns of A. We let A′^ denote the resulting matrix. Obtaining A′^ from A requires i − 1 row interchanges and j − 1 column interchanges. Therefore, the total number of interchanges required to obtain A′^ from A is i + j − 2. Consequently,
det(A) = (− 1 )i+j^ −^2 det(A′) = (− 1 ) i+j^ det(A′).
Now for the key point. The coefficient of aij in det(A) must be (− 1 )i+j^ times the coefficient of aij in det(A′). But, aij occurs in the (1, 1) position of A′, and so, as we have previously shown, its coefficient in det(A′) is M 11 ′. Since the relative positions of the remaining rows in A have not altered, it follows that M 11 ′ = Mij , and therefore the coefficient of aij in det(A′) is Mij. Consequently, the coefficient of aij in det(A) is (− 1 )i+j^ Mij = Cij. Applying this result to the elements ai 1 , ai 2 ,... , a (^) in and comparing with (3.3.1) yields C^ ˆij = Cij , j = 1 , 2 ,... , n,
which establishes the theorem for expansion along a row. The result for expansion along a column follows directly, since det(AT^ ) = det(A).
We now have two computational methods for evaluating determinants: the use of elementary row operations given in the previous section to reduce the matrix in question to upper triangular form, and the Cofactor Expansion Theorem. In evaluating a given determinant by hand, it is usually most efficient (and least error prone) to use a com- bination of the two techniques. More specifically, we use elementary row operations to set all except one element in a row or column equal to zero and then use the Cofactor Expansion Theorem on that row or column. We illustrate with an example.
Example 3.3.10 Evaluate 2 1 8 6 1 4 1 3 −1 2 1 4 1 3 −1 2
Solution: We have
2 1 8 6 1 4 1 3 −1 2 1 4 1 3 −1 2
=^1
= −^2
= −^3
=^4 .
1. A 21 (− 2 ), A 23 ( 1 ), A 24 (− 1 ) 2. Cofactor expansion along column 1 3. A 32 ( 7 ) 4. Cofactor expansion along column 3
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218 CHAPTER 3 Determinants
That is,
det(A) = det(A) +
∑^ n
k= 1
aik Cj k ,
since by the Cofactor Expansion Theorem the first summation on the right-hand side is simply det(A). It follows immediately that
∑^ n
k= 1
aik Cj k = 0 , i = j.
Equation (3.3.3) can be proved similarly (Problem 47).
The Cofactor Expansion Theorem and the above corollary can be combined into the following corollary.
Corollary 3.3.13 Let A be an n × n matrix. If δ (^) ij is the Kronecker delta symbol (see Definition 2.2.19), then
∑^ n
k= 1
aik Cj k = δij det(A),
∑n
k= 1
aki Ckj = δij det(A). (3.3.4)
The formulas in (3.3.4) should be reminiscent of the index form of the matrix product. Combining this with the fact that the Kronecker delta gives the elements of the identity matrix, we might suspect that (3.3.4) is telling us something about the inverse of A. Before establishing that this suspicion is indeed correct, we need a definition.
DEFINITION 3.3.
If every element in an n × n matrix A is replaced by its cofactor, the resulting matrix is called the matrix of cofactors and is denoted MC. The transpose of the matrix of cofactors, M (^) CT , is called the adjoint of A and is denoted adj(A). Thus, the elements of adj(A) are adj(A)ij = Cj i.
Example 3.3.15 Determine adj(A) if
A =
Solution: We first determine the cofactors of A:
C 11 = 8 , C 12 = 12 , C 13 = − 13 , C 21 = 6 , C 22 = 9 , C 23 = 4 , C 31 = 15 , C 32 = − 5 , C 33 = 10.
Thus,
MC =
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3.3 Cofactor Expansions 219
so that
adj(A) = M (^) CT =
We can now prove the next theorem.
Theorem 3.3.16 (The Adjoint Method for Computing A−^1 ) If det(A) = 0, then
A−^1 =
det(A)
adj(A).
Proof Let B =
det(A)
adj(A). Then we must establish that AB = I (^) n = BA. But, using the index form of the matrix product,
(AB)ij =
∑^ n
k= 1
aik bkj =
∑^ n
k= 1
aik ·
det(A)
· adj(A)kj =
det(A)
∑n
k= 1
aik Cj k = δij ,
where we have used Equation (3.3.4) in the last step. Consequently, AB = In. We leave it as an exercise (Problem 53) to verify that BA = In also.
Example 3.3.17 For the matrix in Example 3.3.15,
det(A) = 55 ,
so that
A−^1 =
For square matrices of relatively small size, the adjoint method for computing A−^1 is often easier than using elementary row operations to reduce A to upper triangular form. In Chapter 7, we will find that the solution of a system of differential equations can be expressed naturally in terms of matrix functions. Certain problems will require us to find the inverse of such matrix functions. For 2 × 2 systems, the adjoint method is very quick.
Example 3.3.18 Find A−^1 if A =
[
e^2 t^ e−t 3 e^2 t^6 e−t
]
Solution: In this case,
det(A) = (e^2 t^ )( 6 e−t^ ) − ( 3 e^2 t^ )(e−t^ ) = 3 e t^ ,
and adj(A) =
[
6 e−t^ −e−t − 3 e^2 t^ e^2 t
]
so that
A−^1 =
[
2 e−^2 t^ − 13 e−^2 t −e t^13 e t
]
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3.3 Cofactor Expansions 221
det(A)
∑n
i= 1
Cik bi , k = 1 , 2 ,... , n.
Using (3.3.5), we can write this as
xk = det(Bk ) det(A)
, k = 1 , 2 ,... , n
as required.
Remark In general, Cramer’s rule requires more work than the Gaussian elimination method, and it is restricted to n × n systems whose coefficient matrix is invertible. However, it is a powerful theoretical tool, since it gives us a formula for the solution of an n × n system, provided det(A) = 0.
Example 3.3.20 Solve 3 x 1 + 2 x 2 − x 3 = 4 , x 1 + x 2 − 5 x 3 = − 3 , − 2 x 1 − x 2 + 4 x 3 = 0.
Solution: The following determinants are easily evaluated:
det(A) =
= 8 , det(B 1 ) =
det(B 2 ) =
= − 6 , det(B 3 ) =
Inserting these results into (3.3.6) yields x 1 = 178 , x 2 = − 68 = − 34 , and x 3 = 78 , so that the solution to the system is ( 178 , − 34 , 78 ).
Exercises for 3.
Key Terms
Minor, Cofactor, Cofactor expansion, Matrix of cofactors, Adjoint, Cramer’s rule.
Skills
- Be able to compute the minors and cofactors of a matrix.
- Understand the difference between Mij and Cij.
- Be able to compute the determinant of a matrix via cofactor expansion.
- Be able to compute the matrix of cofactors and the adjoint of a matrix. - Be able to use the adjoint of an invertible matrix A to compute A−^1. - Be able to use Cramer’s rule to solve a linear system of equations.
True-False Review
For Questions 1–7, decide if the given statement is true or false , and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.
1. The ( 2 , 3 )-minor of a matrix is the same as the ( 2 , 3 )- cofactor of the matrix. 2. We have A · adj(A) = det(A) · I (^) n for all n × n matrices A.
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222 CHAPTER 3 Determinants
3. Cofactor expansion of a matrix along any row or col- umn will yield the same result, although the individual terms in the expansion along different rows or columns can vary. 4. If A is an n × n matrix and c is a scalar, then
adj(cA) = c · adj(A).
5. If A and B are 2 × 2 matrices, then
adj(A + B) = adj(A) + adj(B).
6. If A and B are 2 × 2 matrices, then
adj(AB) = adj(A) · adj(B).
7. For every n, adj(In ) = In.
Problems
For Problems 1–3, determine all minors and cofactors of the given matrix.
1. A =
[
]
2. A =
3. A =
4. If
A =
determine the minors M 12 , M 31 , M 23 , M 42 , and the corresponding cofactors.
For Problems 5–10, use the Cofactor Expansion Theorem to evaluate the given determinant along the specified row or column.
, row 1.
, column 3.
, row 2.
, column 1.
, row 3.
, column 4.
For Problems 11–19, evaluate the given determinant using the techniques of this section.
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224 CHAPTER 3 Determinants
38. A =
e t^ te t^ e−^2 t e t^2 te t^ e−^2 t e t^ te t^2 e−^2 t
39. If
A =
compute the matrix product A · adj(A). What can you conclude about det(A)?
For Problems 40–43, use Cramer’s rule to solve the given linear system.
2 x 1 − 3 x 2 = 2 , x 1 + 2 x 2 = 4.
3 x 1 − 2 x 2 + x 3 = 4 , x 1 + x 2 − x 3 = 2 , x 1 + x 3 = 1.
x 1 − 3 x 2 + x 3 = 0 , x 1 + 4 x 2 − x 3 = 0 , 2 x 1 + x 2 − 3 x 3 = 0.
x 1 − 2 x 2 + 3 x 3 − x 4 = 1 , 2 x 1 + x 3 = 2 , x 1 + x 2 − x 4 = 0 , x 2 − 2 x 3 + x 4 = 3.
44. Use Cramer’s rule to determine x 1 and x 2 if
e t^ x 1 + e−^2 t^ x 2 = 3 sin t, e t^ x 1 − 2 e−^2 t^ x 2 = 4 cos t.
45. Determine the value of x 2 such that
x 1 + 4 x 2 − 2 x 3 + x 4 = 2 , 2 x 1 + 9 x 2 − 3 x 3 − 2 x 4 = 5 , x 1 + 5 x 2 + x 3 − x 4 = 3 , 3 x 1 + 14 x 2 + 7 x 3 − 2 x 4 = 6.
46. Find all solutions to the system
(b + c)x 1 + a(x 2 + x 3 ) = a, (c + a)x 1 + b(x 3 + x 1 ) = b, (a + b)x 1 + c(x 1 + x 2 ) = c,
where a, b, c are constants. Make sure you consider all cases (that is, those when there is a unique solution, an infinite number of solutions, and no solutions).
47. Prove Equation (3.3.3). 48. Let A be a randomly generated invertible 4 × 4 ma- trix. Verify the Cofactor Expansion Theorem for ex- pansion along row 1. 49. Let A be a randomly generated 4 × 4 matrix. Verify Equation (3.3.3) when i = 2 and j = 4. 50. Let A be a randomly generated 5 × 5 matrix. Deter- mine adj(A) and compute A · adj(A). Use your result to determine det(A). 51. Solve the system of equations
- 21 x 1 + 3. 42 x 2 + 2. 15 x 3 = 3. 25 ,
- 41 x 1 + 2. 32 x 2 + 7. 15 x 3 = 4. 61 ,
- 63 x 1 + 3. 51 x 2 + 9. 22 x 3 = 9. 93.
Round answers to two decimal places.
52. Use Cramer’s rule to solve the system A x = b if
A =
, and b =
53. Verify that BA = In in the proof of Theorem 3.3.16.
3.4 Summary of Determinants
The primary aim of this section is to serve as a stand-alone introduction to determinants for readers who desire only a cursory review of the major facts pertaining to determinants. It may also be used as a review of the results derived in Sections 3.1–3.3.
Formulas for the Determinant
The determinant of an n × n matrix A, denoted det(A), is a scalar whose value can be obtained in the following manner.
1. If A = [a 11 ], then det(A) = a 11.
