Determinants Lecture 1

A 3xx3 determinant has entries either 1 or -1 . Let S_(3)= set of all determinants which contain determinants such that product of elements of any row or any column is -1 For example |{:(1,,-1,,1),(1,,1,,-1),(-1,,1,,1):}| is an element of the set S_(3) . Number of elements of the set S_(3)=

Consider the set A of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of the set of all determinants with value -1 . Then

A determinant is chosen at random from the set of all determinant of order 2 with elements 0 or 1 only. Find the probability that the determinant chosen is nonzero.

Using properties of determinants, prove the following: |[1,a,a^2],[a^2,1,a],[a,a^2,1]|=(1-a^3)^2

Using properties of determinants, prove that 3 2 (a 1) 3 3 1 2a 1 a 2 1 a 2a 2a

Using properties of determinants, prove that |(a^2+2a, 2a+1,1), (2a+1, a+2, 1), (3,3,1)| = (a-1)^3

Using properties of determinants, prove that |(a^2+2a, 2a+1,1), (2a+1, a+2, 1), (3,3,1)| = (a-1)^3

Entries of a 2 xx 2 determinant are chosen from the set {-1, 1} . The probability that determinant has zero value is

If a determinant of order 3xx3 is formed by using the numbers 1 or -1 then minimum value of determinant is :

A determinant of second order is made with the elements 0 and 1. Find the number of determinants with non-negative values.