and 퐶 be a 2 × 3 real matrix. Which of the following operations are defined?

For those operations that are defined, what is the order of the resulting matrix?

Let R is commutative.

Calculate the following:

(i) (3x² + 4x + 1) + (3x3 + 4x² + 2x + 3) in Z₅[x].

(ii) (3x²+2x+6). (3x3+4x+5) in Z₇[x].

Let R be a ring (not necessarily commutative) and I and J be ideal of R. Show that I   J  and I + J = } are ideals of R.

Define a subgroup of a group. Check whether 

is a subgroup of the the group of 2 × 3 matrices over ℂ under addition

and ∗ be the binary operation defined by  Compute the Cayley table focommutative? Is ∗ associative? Justify your answers.

Show that, if G is a finite group and Further, show that    Deduce the Euler-Fermat theorem