Linear Algebra/Laplace's Expansion/Solutions

Solutions Problem 1 Find the cofactor. <li> <li> <li> Answer <li> <li> <li> Problem 2 Find the determinant by expanding <li> on the first row <li> on the second row <li> on the third column. Answer <li> <li> <li> Problem 3 Find the adjoint of the matrix in Example 1.6. Answer Problem 4 Find the matrix adjoint to each. <li> <li> <li> <li> Answer <li> <li> The minors are : . <li> <li> Problem 5 Find the inverse of each matrix in the prior question with Theorem 1.9. Answer <li> <li> <li> <li> The matrix has a zero determinant, and so has no inverse. Problem 6 Find the matrix adjoint to this one. Answer Problem 7 Expand across the first row to derive the formula for the determinant of a matrix. Answer The determinant expanded on the first row gives (note the two minors). Problem 8 Expand across the first row to derive the formula for the determinant of a matrix. Answer The determinant of is this. Problem 9 <li> Give a formula for the adjoint of a matrix. <li> Use it to derive the formula for the inverse. Answer <li> <li> Problem 10 Can we compute a determinant by expanding down the diagonal? Answer No. Here is a determinant whose value doesn't equal the result of expanding down the diagonal. Problem 11 Give a formula for the adjoint of a diagonal matrix. Answer Consider this diagonal matrix. If then the minor is an matrix with only nonzero entries, because both and are deleted. Thus, at least one row or column of the minor is all zeroes, and so the cofactor is zero. If then the minor is the diagonal matrix with entries , ..., , , ..., . Its determinant is obviously times the product of those. By the way, Theorem 1.9 provides a slicker way to derive this conclusion. Problem 12 Prove that the transpose of the adjoint is the adjoint of the transpose. Answer Just note that if then the cofactor equals the cofactor because and because the minors are the transposes of each other (and the determinant of a transpose equals the determinant of the matrix). Problem 13 Prove or disprove: . Answer It is false; here is an example. Problem 14triangular A square matrix is upper triangular if each entry is zero in the part above the diagonal, that is, when . <li> Must the adjoint of an upper triangular matrix be upper triangular? Lower triangular? <li> Prove that the inverse of a upper triangular matrix is upper triangular, if an inverse exists. Answer <li> An example suggests the right answer. The result is indeed upper triangular. A check of this is detailed but not hard. The entries in the upper triangle of the adjoint are where . We need to verify that the cofactor is zero if . With , row and column of , when deleted, leave an upper triangular minor, because entry of the minor is either entry of (this happens if and ; in this case implies that the entry is zero) or it is entry of (this happens if and ; in this case, implies that , which implies that the entry is zero), or it is entry of (this last case happens when and ; obviously here implies that and so the entry is zero). Thus the determinant of the minor is the product down the diagonal. Observe that the entry of is the entry of the minor (it doesn't get deleted because the relation is strict). But this entry is zero because is upper triangular and . Therefore the cofactor is zero, and the adjoint is upper triangular. (The lower triangular case is similar.) <li> This is immediate from the prior part, by Corollary 1.11. Problem 15exer:LaplaceProof This question requires material from the optional Determinants Exist subsection. Prove Theorem 1.5 by using the permutation expansion. Answer We will show that each determinant can be expanded along row . The argument for column is similar. Each term in the permutation expansion contains one and only one entry from each row. As in Example 1.1, factor out each row entry to get , where each is a sum of terms not containing any elements of row . We will show that is the cofactor. Consider the case first: where the sum is over all -permutations such that . To show that is the minor , we need only show that if is an -permutation such that and is an -permutation with , ..., then . But that's true because and have the same number of inversions. Back to the general case. Swap adjacent rows until the -th is last and swap adjacent columns until the -th is last. Observe that the determinant of the -th minor is not affected by these adjacent swaps because inversions are preserved (since the minor has the -th row and -th column omitted). On the other hand, the sign of and is changed plus times. Thus . Problem 16 Prove that the determinant of a matrix equals the determinant of its transpose using Laplace's expansion and induction on the size of the matrix. Answer This is obvious for the base case. For the inductive case, assume that the determinant of a matrix equals the determinant of its transpose for all , ..., matrices. Expanding on row gives and expanding on column gives Since the signs are the same in the two summations. Since the minor of is the transpose of the minor of , the inductive hypothesis gives . ? Problem 17 Show that where is the -th term of , the Fibonacci sequence, and the determinant is of order . Walter Tytun 1949 Answer This is how the answer was given in the cited source. Denoting the above determinant by , it is seen that , . It remains to show that . In subtract the -th column from the -th, the -th from the -th, ..., the first from the third, obtaining By expanding this determinant with reference to the first row, there results the desired relation. References Dan (proposer) Walter Alex (solver) Tytun Elementary problem 834 American Mathematical Monthly American Mathematical Society 56 6 409 1949.