Differential Equations and Linear Algebra C. Henry Edwards David E. Penney Third Edition

Transcription

1 Differential Equations and Linear Algebra C. Henry Edwards David E. Penney Third Edition

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3 204 Chapter 3 Linear Systems and Matrices DEFINITION Minors and Cofactors Let A = [ a ij ] be an n n matrix. The ijth minor of A (also called the minor of a ij ) is the determinant M ij of the (n 1) (n 1) submatrix that remains after deleting the ith row and the jth column of A. The ijth cofactor A ij of A (or the cofactor of a ij ) is defined to be A ij = ( 1) i+ j M ij. (6) For example, the minor of a 12 ina3 3 matrix is a 11 a 12 a 13 M 12 = a 21 a 22 a 23 a 31 a 32 a 33 The minor of a 32 ina4 4 matrix is a 11 a 12 a 13 a 14 a M 32 = 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 = a 21 a 23. a 31 a 33 = a 11 a 13 a 14 a 21 a 23 a 24 a 41 a 43 a 44 According to Eq. (6), the cofactor A ij is obtained by attaching the sign ( 1) i+ j to the minor M ij. This sign is most easily remembered as the one that appears in the ijth position in checkerboard arrays such as and Note that a plus sign always appears in the upper left corner and that the signs alternate horizontally and vertically. In the 4 4 case, for instance, A 11 =+M 11, A 12 = M 12, A 13 =+M 13, A 14 = M 14, A 21 = M 21, A 22 =+M 22, A 23 = M 23, A 24 =+M 24, and so forth. With this notation, the definition of 3 3 determinants in (5) can be rewritten as det A = a 11 M 11 a 12 M 12 + a 13 M 13 = a 11 A 11 + a 12 A 12 + a 13 A 13. (7) The last formula is the cofactor expansion of det A along the first row of A. Its natural generalization yields the definition of the determinant of an n n matrix, under the inductive assumption that (n 1) (n 1) determinants have already been defined. 204

4 3.6 Determinants 205 DEFINITION n n Determinants The determinant det A = a ij of an n n matrix A = [ a ij ] is defined as det A = a 11 A 11 + a 12 A a 1n A 1n. (8) Thus we multiply each element of the first row of A by its cofactor and then add these n products to get det A. In numerical computations, it frequently is more convenient to work first with minors rather than with cofactors and then attach signs in accord with the checkerboard pattern illustrated previously. Note that determinants have been defined only for square matrices. Example 4 To evaluate the determinant of A = , we observe that there are only two nonzero terms in the cofactor expansion along the first row. We need not compute the cofactors of zeros, because they will be multiplied by zero in computing the determinant; hence det A =+(2) ( 3) Each of the 3 3 determinants on the right-hand side has only a single nonzero term in its cofactor expansion along the first row, so det A = (2)( 1) (3)(+1) = ( 2)(12 10) + (3)( ) = 92. Note that, if we could expand along the second row in Example 4, there would be only a single 3 3 determinant to evaluate. It is in fact true that a determinant can be evaluated by expansion along any row or column. The proof of the following theorem is included in Appendix B.. THEOREM 1 Cofactor Expansions of Determinants The determinant of an n n matrix A = [ a ij ] can be obtained by expansion along any row or column. The cofactor expansion along the ith row is det A = a i1 A i1 + a i2 A i2 + +a in A in. (9) The cofactor expansion along the jth column is det A = a 1 j A 1 j + a 2 j A 2 j + +a nj A nj. (10) 205

5 206 Chapter 3 Linear Systems and Matrices The formulas in (9) and (10) provide 2n different cofactor expansions of an n n determinant. For n = 3, for instance, we have det A = a 11 A 11 + a 12 A 12 + a 13 A 13 = a 21 A 21 + a 22 A 22 + a 23 A 23 row expansions = a 31 A 31 + a 32 A 32 + a 33 A 33 = a 11 A 11 + a 21 A 21 + a 31 A 31 = a 12 A 12 + a 22 A 22 + a 32 A 32 column expansions = a 13 A 13 + a 23 A 23 + a 33 A 33. In a specific example, we naturally attempt to choose the expansion that requires the least computational labor. Example 5 To evaluate the determinant of A = , we expand along the third column, because it has only a single nonzero entry. Thus det A = (2) = ( 2)(35 24) = 22. In addition to providing ways of evaluating determinants, the theorem on cofactor expansions is a valuable tool for investigating the general properties of determinants. For instance, it follows immediately from the formulas in (9) and (10) that, if the square matrix A has either an all-zero row or an all-zero column, then det A = 0. For example, by expanding along the second row we see immediately that Row and Column Properties = 0. We now list seven properties of determinants that simplify their computation. Each of these properties is readily verified directly in the case of 2 2 determinants. Just as our definition of n n determinants was inductive, the following discussion of Properties 1 7 of determinants is inductive. That is, we suppose that n 3 and that these properties have already been verified for (n 1) (n 1) determinants. PROPERTY 1: If the n n matrix B is obtained from A by multiplying a single row (or a column) of A by the constant k, then det B = k det A. For instance, if the ith row of A is multiplied by k, then the elements off the ith row of A are unchanged. Hence for each j = 1, 2,...,n, the ijth cofactors of A and B are equal: A ij = B ij. Therefore, expansion of B along the ith row gives and thus det B = k det A. det B = (ka i1 )B i1 + (ka i2 )B i2 + +(ka in )B in = k(a i1 A i1 + a i2 A i2 + +a in A in ), 206

6 3.6 Determinants 207 Property 1 implies simply that a constant can be factored out of a single row or column of a determinant. Thus we see that = (3) by factoring 3 out of the second column. PROPERTY 2: If the n n matrix B is obtained from A by interchanging two rows (or two columns), then det B = det A. To see why this is so, suppose (for instance) that the first row is not one of the two that are interchanged (recall that n 3). Then for each j = 1, 2,...,n, the cofactor B 1 j is obtained by interchanging two rows of the cofactor A 1 j. Therefore, B 1 j = A 1 j by Property 2 for (n 1) (n 1) determinants. Because b 1 j = a 1 j for each j, it follows by expanding along the first row that and thus det B = det A. det B = b 11 B 11 + b 12 B b 1n B 1n = a 11 ( A 11 ) + a 12 ( A 12 ) + +a 1n ( A 1n ) = (a 11 A 11 + a 12 A a 1n A 1n ), PROPERTY 3: If two rows (or two columns) of the n n matrix A are identical, then det A = 0. To see why, let B denote the matrix obtained by interchanging the two identical rows of A. Then B = A, so det B = det A. But Property 2 implies that det B = det A. Thus det A = det A, and it follows immediately that det A = 0. PROPERTY 4: Suppose that the n n matrices A 1, A 2, and B are identical except for their ith rows that is, the other n 1 rows of the three matrices are identical and that the ith row of B is the sum of the ith rows of A 1 and A 2. Then det B = det A 1 + det A 2. This result also holds if columns are involved instead of rows. Property 4 is readily established by expanding B along its ith row. In Problem 45 we ask you to supply the details for a typical case. The main importance (at this point) of Property 4 is that it implies the following property relating determinants and elementary row operations. PROPERTY 5: If the n n matrix B is obtained by adding a constant multiple of one row (or column) of A to another row (or column) of A, then det B = det A. Thus the value of a determinant is not changed either by the type of elementary row operation described or by the corresponding type of elementary column operation. The following computation with 3 3 matrices illustrates the general proof of Property 5. Let A = a 11 a 12 a 13 a 21 a 22 a 23 and B = a 11 a 12 a 13 + ka 11 a 21 a 22 a 23 + ka 21. a 31 a 32 a 33 a 31 a 32 a 33 + ka

7 208 Chapter 3 Linear Systems and Matrices So B is the result of adding k times the first column of A to its third column. Then a 11 a 12 a 13 + ka 11 det B = a 21 a 22 a 23 + ka 21 a 31 a 32 a 33 + ka 31 a 11 a 12 a 13 = a 21 a 22 a 23 a 31 a 32 a 33 + k a 11 a 12 a 11 a 21 a 22 a 21 a 31 a 32 a 31. (11) Here we first applied Property 4 and then factored k out of the second summand with the aid of Property 1. Now the first determinant on the right-hand side in (11) is simply det A, whereas the second determinant is zero (by Property 3 its first and third columns are identical). We therefore have shown that det B = det A, as desired. Although we use only elementary row operations in reducing a matrix to echelon form, Properties 1, 2, and 5 imply that we may use both elementary row operations and the analogous elementary column operations in simplifying the evaluation of determinants. Example 6 The matrix A = has no zero elements to simplify the computation of its determinant as it stands, but we notice that we can knock out the first two elements of its third column by adding twice the first column to the third column. This gives det A = = = (+7) = 35. The moral of the example is this: Evaluate determinants with your eyes open. An upper triangular matrix is a square matrix having only zeros below its main diagonal. A lower triangular matrix is a square matrix having only zeros above its main diagonal. A triangular matrix is one that is either upper triangular or lower triangular, and thus looks like or The next property tells us that determinants of triangular matrices are especially easy to evaluate. 208