## The definition of

Let denote a field and let denote the set of all matrices with entries in having non-zero determinant. This is called the general linear group of degree over . Each element defined a function by (5.1).

Proposition 5.5.4   is a group under ordinary matrix multiplication.

proof: The identity matrix is the identity element. Associativity is a property of matrix multiplication. (Left to the reader as an exercise.) The set is closed under multiplication by Lemma 5.5.3. Inverses exist by Lemma 5.5.2.

Let

be a matrix with . Suppose that the determinant 5.3 of is , . Thanks to Theorem 1.4.3, this forces (why?). Similarly, we must have , , and . On other words, to each integer matrix with determinant , is associated several pairs of integers with no common factor.

Conversely, if have no common factor then by Theorem 1.4.3 there are such that . This means that the matrix

has determinant .

Exercise 5.5.5   Let be the subset of all such that . Show that is a group.

Exercise 5.5.6   Let be the subset of all such that . Show that is a group.

Exercise 5.5.7   Find two matrices of the form

having determinant .

Exercise 5.5.8   Find a matrix of the form

having determinant .

Exercise 5.5.9   Find a matrix of the form

having determinant .

Exercise 5.5.10   Find a matrix of the form

having determinant .

Exercise 5.5.11   Compute

using the Lagrange expansion

Exercise 5.5.12   Let

have determinant . Show that there is a matrix

with and having determinant , such that has the form

for some . (Hint: Take , , and use Theorem 1.4.3 to determine .)

Exercise 5.5.13   Imagine a chessboard in front of you. You can place at most non-attacking rooks on the chessboard. (Rooks move only horizontally and vertically.) Now imagine you have done this and let be the matrix of 0 's and 's (called a -matrix) where if there is a rook on the square belonging to the horizontal down and the vertical from the left. Call such a matrix a rook matrix. If there are exactly 8 's in then we shall call a full rook matrix. For example,

is a full rook matrix. Show that

(a) a full rook matrix is an permutation matrix,

(b) any full rook matrix is invertible,

(c) the product of any two rook matrices is a rook matrix.

david joyner 2008-04-20