If `(AB)^(-1)=A^(-1)B^(-1)`, then prove that `A^(-1)` and `B^(-1)` satisfy commutative property with respect to multiplication.

Three matrices A,Band C defined as A=[{:(1,2),(-1,3):}],B=[{:(4,0),(1,5):}]andC=[{:(2,0),(1,-2):}] , then show that matrices A,B and C satisfy (i) the property of associativity with respect to multiplication ,i.e A(BC)=(AB)C. (ii) the property of distributivity with respect to addition i.e A(B+C)=AB+AC.

If "tan"^(-1)a+"tan"^(-1)b+"tan"^(-1)c=pi , then prove that a+b+c=abc .

If abs((a,b,a-b),(b,c,b-c),(2,1,0))=0 , then prove that a,b,c are in GP.

If (a-c)^2,(c-a)^2,(a-b)^2 are in A.P.,prove that 1/(b-c),1/(c-a),1/(a-b) are in A.P.

Let A=[(3,7),(2,5):}] and B=[(6,8),(7,9):}] then verify that (AB)^-1 =B^-1 A-1

Prove that A-oversetn underset(i=1) uuu B_i =overset n underset (i=1) nnn (A -B_i)

If A=[{:(1,-2,3),(-4,2,5):}] and B=[{:(2,3),(4,5),(2,1):}] then find AB and BA. Show that A and B are not commutative, i.e. AB ne BA .