Existence of identity; existence of inverse and cancellation laws

Define an operation * on Q as follows: a * b =((a+b)/(2)), a, b in Q . Examine the existence of identify and existence of inverse for the operation * on Q.

(i) Let M={({:(x,x),(x,x):}):x in R -{0}} and let ** be the matrix multiplication. Determine whether M is closed under ** . If so, examinie the existence of identity, existence of inverse properties for the operation ** on M.

Verify (i) closure property (ii) commutative property (iii) associative property (iv) existence of identity and (v) existence of inverse for the operation +_(5) on ZZ_5 using table corresponding to addition modulo 5.

Existence of additive identity

Existence of additive inverse

Existence of negative (additive inverse) of a rational number

Examine the existence of inverse matrix and find the inverse matrix I it exists : A=[(1,2),(2,4)]