Linear Algebra

Section I. Definition 2972.20 Prove that the determinant of a product is the product of the determinants|T S| = |T | |S| in this way. Fix the n × n matrix S and consider the functiond: M n×n → R given by T ↦→ |T S|/|S|.(a) Check that d satisfies property (1) in the definition of a determinant function.(b) Check property (2).(c) Check property (3).(d) Check property (4).(e) Conclude the determinant of a product is the product of the determinants.2.21 A submatrix of a given matrix A is one that can be obtained by deleting someof the rows and columns of A. Thus, the first matrix here is a submatrix of thesecond.( ) ( )3 4 13 10 9 −22 52 −1 5Prove that for any square matrix, the rank of the matrix is r if and only if r is thelargest integer such that there is an r×r submatrix with a nonzero determinant.̌ 2.22 Prove that a matrix with rational entries has a rational determinant.? 2.23 Find the element of likeness in (a) simplifying a fraction, (b) powdering thenose, (c) building new steps on the church, (d) keeping emeritus professors oncampus, (e) putting B, C, D in the determinant1 a a 2 a 3a 3 1 a a 2B a 3 .1 a∣C D a 3 1 ∣[Am. Math. Mon., Feb. 1953]I.3 The Permutation ExpansionThe prior subsection defines a function to be a determinant if it satisfies fourconditions and shows that there is at most one n×n determinant function foreach n. What is left is to show that for each n such a function exists.How could such a function not exist? After all, we have done computationsthat start with a square matrix, follow the conditions, and end with a number.The difficulty is that, as far as we know, the computation might not give awell-defined result. To illustrate this possibility, suppose that we were to changethe second condition in the definition of determinant to be that the value of adeterminant does not change on a row swap. By Remark 2.2 we know thatthis conflicts with the first and third conditions. Here is an instance of theconflict: here are two Gauss’ method reductions of the same matrix, the firstwithout any row swap( ) ( )1 2 −3ρ 1+ρ 2 1 2−→3 40 −2