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8. The of a is either 1 or −1 depending on whether it takes an even or an odd number of row in- terchanges to convert it to the .

Determinants, part III Other properties of . There are Math 130 Linear several other important properties of determinants. D Joyce, Fall 2015 For instance, determinants can be evaluated by cofactor expansion along any row, not just the first We characterized what determinant functions are row as we used to construct the determinant. We based on four properties, and we saw one construc- won’t take the time to prove that. The idea of tion for them. Here’s a quick summary of their the proof is straightforward—exchange the given properties. The first four characterize them; the row with the first row and apply cofactor expan- others we proved. sion along the first row. The only difficultly with the proof is keeping track of the of the deter- A determinant assigns to each square minant. matrix A a associated to the matrix, denoted det(A) or |A| such that As mentioned before, we won’t use cofactor ex- pansion much since it’s not a practical way to eval- 1. The determinant of an n × n identity matrix I uate determinants. Here are a couple of more useful is 1. |I| = 1. properties.

2. If the matrix B is identical to the matrix A Theorem 1. If one row of a is a except the entries in one of the rows of B are multiple of another row, then its determinant is 0. each equal to the corresponding entries of A multiplied by the same scalar , then |B| = Proof. We saw that if two rows are the same, then c |A|. a square matrix has 0 determinant. By the second property of determinants if we multiply one of those 3. If the matrices A, B, and C are identical except rows by a scalar, the matrix’s determinant, which is for the entries in one row, and for that row an 0, is multiplied by that scalar, so that determinant entry in A is found by adding the correspond- is also 0. q.e.d. ing entries in B and C, then |A| = |B| + |C|. Theorem 2. The determinant of a matrix is not 4. If the matrix B is the result of exchanging two changed when a multiple of one row is added to rows of A, then the determinant of B is the another. negation of the determinant of A.

5. The determinant of any matrix with an entire Proof. Let A be the given matrix, and let B be the row of 0’s is 0. matrix that results if you add c times row k to row l, k 6= l. Let C be the matrix that looks just like th th 6. The determinant of any matrix with two iden- A except the l row of C is c times the k row. tical rows is 0. Since one row of C is a multiple of another row of C, its determinant is 0. By multilinearity of the 7. There is one and only one determinant func- determinant (property 3), |A| = |B| + |C|. Since tion. |C| = 0, therefore |A| = |B|. q.e.d.

1 An efficient for evaluating a ma- the of the of two matrices is less than trix. The three row operations are (1) exchange or equal to the rank of each, therefore the rank of rows, and that will negate the determinant, (2) AB is less than n, and, hence |AB| = 0. Thus, no multiply or divide a row by a nonzero constant, matter what B is, |AB| = |A| |B|. and that scales the determinant by that constant, Case 2. For this case assume the rank of A is n. and (3) add a multiple of one row to another, and Then A can be expressed as a product of elementary that doesn’t change the determinant at all. matrices A = E1E2 ··· Ek. If we knew for each After you’ve row reduced the matrix, you’ll have E that |EB| = |E| |B|, then it a , and its determinant will be the would follow that product of its diagonal entries.

We’ll evaluate a couple of matrices by this |AB| = |E1E2 ··· EkB| method in class. = |E1| |E2| · · · |Ek| |B| = |A| |B| Determinants, rank, and invertibility. There’s a close connection between these for a Thus, we can reduce case 2 to the special case where square matrix. We’ve seen that an n × n matrix A A is an elementary matrix. has an inverse if and only if rank(A) = n. We can Elementary subcases. We’ll show that for each ele- add another equivalent condition to that, namely, mentary matrix E that |EB| = |E| |B|. There are |A|= 6 0. three kinds of elementary matrices, and we’ll check Theorem 3. The determinant of an n × n matrix each one. is nonzero if and only if its rank is n, that is to say, Elementary subcase 1. Suppose that E is the result if and only if it’s invertible. of interchanging two rows of the identity matrix I. Then |E| = −1, and EB is the same as B except Proof. The determinant of the matrix will be 0 if those two rows are interchanged, so |EB| = −|EB|. and only if when it’s row reduced the resulting ma- Thus, |EB| = |E| |B|. trix has a row of 0s, and that happens when its rank is less than n. q.e.d. Elementary subcase 2. Suppose that E is the result of multiplying a row of I by a scalar c. Then |E| = Determinant of products and inverses. c, and EB is the same as B except that row is These are rather important properties of determi- multiplied by c, so |EB| = c |B|. Thus, |EB| = nants. The first says |AB| = |A| |B|, and the sec- |E| |B|. ond says |A−1| = |A|−1 when A is an invertible Elementary subcase 3. Suppose that E is the result matrix. of adding a multiple of one row of I to another. Then |E| = 1, and EB is the same as B except Theorem 4. The determinant of the product of that that same multiple of one row of B is added two square matrices is the product of their deter- the same other of B, so |EB| = |B|. Again, |EB| = minants, that is, |AB| = |A| |B|. |E| |B|. q.e.d. Proof. We’ll prove this in two cases, first when A Corollary 5. For an , the deter- has rank less than n, then when A has full rank. minant of its inverse is the reciprocal of its deter- For the full rank case we’ll reduce the proof to the minant, that is, |A−1| = |A|−1. case where A is an elementary case since it’s easy to show the result in that case. Proof. According to the theorem, |A−1A| = Case 1. Assume that the rank of A is less than |A| |A−1|, but |A−1A| = |I| = 1, so |A| |A−1| = 1 n. Then by the previous theorem, |A| = 0. Since from which it follows that |A−1| = |A|−1. q.e.d.

2 More generally, |Ap| = |A|p for general p even The following properties of determinants that re- when p is negative so long as A is an invertible late to the columns of a matrix follow from this matrix. theorem and the corresponding properties for rows of a matrix.

Determinants and transposes. So far, every- 1. If the matrix B is identical to the matrix A thing we’ve said about determinants of matrices except the entries in one of the columns of B was related to the rows of the matrix, so it’s some- are each equal to the corresponding entries of what surprising that a matrix and its A multiplied by the same scalar c, then |B| = have the same determinant. We’ll prove that, and c |A|. from that theorem we’ll automatically get corre- sponding statements for columns of matrices that 2. If the matrices A, B, and C are identical ex- we have for rows of matrices. cept for the entries in one column, and for that column an entry in A is found by adding Theorem 6. The determinant of the transpose of the corresponding entries in B and C, then a square matrix is equal to the determinant of the |A| = |B| + |C|. matrix, that is, |At| = |A|. 3. If the matrix B is the result of exchanging two Proof. We’ll prove this like the last theorem. First columns of A, then the determinant of B is the in the case where the rank of A is less than n, then negation of the determinant of A. the case where the rank of A is n, and for the sec- ond case we’ll write A as a product of elementary 4. The determinant of any matrix with an entire matrices. column of 0’s is 0. Case 1. Assume that the rank of A is less than n. 5. The determinant of any matrix with two iden- Then its determinant is 0. But the rank of a matrix tical columns is 0. is the same as the rank of its transpose, so At has rank less than n and its determinant is also 0. 6. The determinant of a is Case 2. For this case assume the rank of A is n. either 1 or −1 depending on whether it takes Express A as a product of elementary matrices, A = an even number or an odd number of column interchanges to convert it to the identity ma- E1E2 ··· Ek. If we knew for each elementary matrix E that |Et| = |E|, then it would follow that trix. 7. The determinant of a matrix can be evaluated |A| = |E1E2 ··· Ek| by cofactor expansion along any column. = |E1| |E2| · · · |Ek| t t t = |E1| |E2| · · · |Ek| Cramer’s rule. This is a method based on de- t t t t = |E1E2 ··· Ek| = |A | terminants to find the solution to a system of n in n unknowns when there is exactly one (Note how we used the property that the transpose solution. The solution is has the determinant in the of a product equals the product of the traposes.) denominator, and the only time the determinant is Thus, we can reduce case 2 to the special case not zero is when there’s a unique solution. where A is an elementary matrix. The details that Cramer’s rule is one of the oldest applications of |Et| = |E| for each of the three kinds of elementary determinants. It’s not an efficient method to solve matrices are omitted here since that’s easy to verify. a system since row is faster, but it’s an q.e.d. interesting use of determinants.

3 Here’s an example to show how to apply Cramer’s rule. Let’s suppose we have the follow- ing system of three equations in three unknowns.

x + y + 3z = 6 2x + 3y − 4z = −2 3x − 2y + 5z = 7

First, compute the determinant ∆ of the 3 × 3 co- efficient matrix.

1 1 3

∆ = 2 3 −4 = −54

3 −2 5 Next, replace the first column by the constant vec- tor, and compute that determinant.

6 1 3

∆x = −2 3 −4 = −27

7 −2 5

1 Then in the unique solution, x = ∆x/∆ = 2 . Next, replace the second column by the constant vector, and compute that determinant.

1 6 3

∆y = 2 −2 −4 = −54

3 7 5

So y = ∆y/∆ = 1. Likewise, replace the third column by the constant vector.

1 1 6

∆z = 2 3 −2 = −81

3 −2 7

3 which gives z = 2 . Thus, the unique solution is 1 3 (x, y, z) = ( 2 , 1, 2 ). Math 130 Home Page at http://math.clarku.edu/~ma130/

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