1.Given two nonsingular matrices A, B elements of Mn(R), show that?

Question

cont..
1.Given two nonsingular matrices A, B elements of Mn(R), show that

A^-1 – (A+B)^-1 = A^-1 (A^-1+ B^-1)^-1 A^-1



5. Definition. Let C element of Mn(R). C is said to be symmetric iff A=Atranspose. Given symmetric matrices A,B elements of Msubscript2(R), show that AB is also symmetric iff A,B commute with each other e.i. AB=BA.

6. If c is an associated eigenvalue of A element of Mn(R), show that c^k is an associated eigenvalue of A^k for some k element of N.

7. If A element of Mn(R) is a nilpotent matrix, show that the only eigenvalue of A is zero.

8. Let A, B elements of Mn(R) such that Ax=cx and Bx=bx. Show that

i.(A+B)x = (c+b)x
ii.(AB)x = (cb)x

Answer 1

8. Let A, B elements of Mn(R) such that Ax=cx and Bx=bx. Show that

i.(A+B)x = (c+b)x
(A+B)x = Ax + Bx = cx + bx =(c+b) x

ii.(AB)x = (cb)x
(AB)x = A(Bx) = A(bx)= bAx=bcx =(cb) x

Answer 2

1. (Supposing that A+B and A^-1+B^-1 are both invertible!)
A^-1 - (A+B)^-1 = A^-1 (A^-1+ B^-1)^-1 A^-1
<=> I - A (A+B)^-1 = (A^-1 + B^-1)^-1 A^-1 (mult on left by A)
<=> (A+B) - A = (A^-1 + B^-1)^-1 A^-1 (A + B) (mult on right by A+B)
<=> (A^-1 + B^-1) (B) = A^-1 (A+B) (simplify LHS and mult on left by A^-1 + B^-1)
<=> A^-1 B + I = I + A^-1 B
which is true.

5. (AB)T = (BA)T = ATBT = AB.

6. If c is an eigenvalue then there is a vector x such that Ax = cx.
Want to prove that A^k x = c^k x for any k in N. True for k=1 by definition, so suppose it's true for k and prove for k+1:
A^(k+1) x = A. A^k x = A (c^k x) = c^k (Ax) = c^k (cx) = c^(k+1) x. So true for all k by induction.

7. Let c be an eigenvalue of A and x an associated eigenvector. Let A^k = 0, then by the last problem c^k x = A^k x = 0x = 0, so c^k = 0 and hence c = 0.

8. i) (A+B) x = Ax + Bx = cx + bx = (c+b)x
ii) (AB)x = A(Bx) = A(bx) = b(Ax) = b(cx) = (bc) x = (cb) x.

Answer 3

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Answer 4

Here's a really handy trick: The set of diagonalizeable matrices is dense in the space of nxn matrices.

Sometimes this observation allows you to reduce the proof to proving it for diagonal matrices.

I haven't checked whether that trick, or some variation, works here, but you could give it a try.