Prove that set of 2 × 2 matrioes form a group under addition

Sure, here is the step-by-step proof that the set of 2x2 matrices forms a group under addition:

1. Closure: If A and B are two 2x2 matrices, then their sum C = A + B is also a 2x2 matrix. Therefore, the set of 2x2 matrices is closed under addition.

2. Associativity: If A, B, and C are three 2x2 matrices, then (A + B) + C = A + (B + C). Therefore, the set of 2x2 matrices is associative under addition.

3. Identity element: The 2x2 zero matrix, denoted by 0, serves as the identity element in this group. For any 2x2 matrix A, A + 0 = A and 0 + A = A.

4. Inverse element: For every 2x2 matrix A, there exists a matrix -A such that A + (-A) = 0 and (-A) + A = 0. Therefore, every element in the set of 2x2 matrices has an inverse under addition.

Since the set of 2x2 matrices satisfies all the four properties (closure, associativity, identity element, and inverse element) under addition, it forms a group under addition.