Verify `(AB)^(-1)=B^(-1)A^(-1)` with `A=[{:(0,-3),(1,4)],B=[(-2,-3),(0,-1):}]`

Verify (AB)^(-1)= B^(-1) A^(-1) for A = [(2,1),(5,3)] and B= [(4,5),(3,4)] .

Verify that (AB)^(T) = B^(T) A^(T) if A= [(2,3,-1),(4,1,5)] and B= [(1,-2),(3,-3),(2,6)]

Verify that (AB)^(T) = B^(T) A^(T) if A = ({:( 2 , 3 , -1) , (4 , 1 , 5):}) and B = ({:( 1 , 2) , ( 3 , -3) , (2 , 6)):}

Compute the indicated products: (i) [(a,b),(-b,a)][(a,-b),(b,a)] (ii) [(1),(2),(3)][(2,3,4)] (iii) [(1,-2),(2,3)][(1,2,3),(2,3,1)] (iv) [(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4),(3,0,5)] (v) [(2,1),(3,2),(-1,1)][(1,0,1),(-1,2,1)] (vi) [(3,-1,3),(-1,0,2)][(2,-3),(1,0),(3,1)]

For the matrices A and B, verify that (AB)'=B'A' , where (i) A=[(1),(-4),(3)], B=[(-1,2,1)] (ii)A=[(0),(1),(2)],B=[(1,5,7)]

Verify that det (AB) = (det A) (det B) for A=[(4,3,-2),(1,0,7),(2,3,-5)] and B =[(1,3,3),(-2,4,0),(9,7,5)] .

Let A+2B=[(1,2,0),(6,-1,3),(-5,3,1)] and 2A-B=[(2,-1,5),(2,-1,6),(0,1,2)], then find tr(A)-tr(B).

Verify the property A(B+C) = AB+AC, when the matrices A,B, and C are given by A=[(2,0,-3),(1,4,5)],B=[(3,1),(-1,0),(4,2)],and C=[(4,7),(2,1),(1,-1)] .