Question a and b please Define Fq 0 1 2 q 1 which has man

Question (a) and (b) please

Define F_q = {0, 1, 2, .., q - 1} which has many, many uses in science. For example, F_2 is the set of binary digits (0, 1) used in computer science. We are going to count how many N x N invertible matrices we can build with elements from the set F4 Note that for this question we are operating over a finite field, so if you want to explicitly calculate the matrices and determinants then you need to do the calculations mod g. (a) What is the cardinality of Fa? (How many elements are there in the set?) (b) We start building the matrix with the first column. Theres only one choice of first column that will definitely make our matrix singular what is it? (c) So how many possible choices do we have for the first column, such that the final matrix still be invertible? (d) Once we have chosen our first column, how many choices of the second column will make the matrix singular? (e) So how many possible choices do we have for the second column, such that the final matrix will still be invertible? once we have chosen the first and second columns, how many choices of the third column will make the matrix singular? (g) Inferring from the previous parts, how many choices are there for the jth column such that the matrix is still invertible? (h) Hence, show that the number of invertible matrices with entries from Fe is (i) How many invertible 2 x 2 matrices with binary digits are there? List them.

Solution

F_q is the set consisting (0,1,2,......,q-1)

Where q is the 1st element .

1 is the second element...

And q-1 is the qth element

Therefore. The cardinality of the set is q

Now . We know that every matrix having 0\'s in a column is singular

i.e; the det of matrix is zero.

Therefore making the first column of n x n matrix to 0\'s the matrix becomes singular.