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(Solved) 5. True or False? For any invertible matrices A = A and B = B , (A + B)" = A" + B". Justify. x): 6. A) Prove that if A = A...


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5. True or False? For any invertible matrices A = A and B = B , (A + B)" = A" + B". Justify. fix): 6. A) Prove that if A = A and B: B are invertible, then AB is also invertible. HXH IXH (Hint: AB multiplied by what matrix gives the identity matrix?)
B) Prove that if A = A and B = B are such that AB is invertible, then each of A and B is It! nxll invertible. ’i'. Let k be a positive integer and let A be an invertible matrix. Use mathematical induction to
prove that (A")" = (A")*. 0 l 2 3
8.LetA= 4 5 6 5 .Find
4 3 2 l A) Elem(R| 6* R3) ' A a) Elem(—%Rz) - A C) Elem(R| — $113) - A
(assume of the indicated elementary matrices have adequate dimensions) (HW 2 continues 011 next page) 1 2
9. Let A = [3 ]. Find an elementary matrix E such that EA = i: ‘3} [-32 :1, [1° ii ['3 i]-

 


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