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Math Help - help with proving nonsingular

  1. #1
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    help with proving nonsingular

    (1) if A and B are nonsingular matrics, which, if any of the following must be nonsingular?

    (a) A+B (b) (AB)^2 (c) kA (k subeset R)

    justify your answer either be obtaining the inverse or by a counter example to show it need not exist.


    (2) if A,B and C are square matrices, such that AB = In and CA = In, prove that A is nonsigular and B = C = A^-1.
    (Hint: try multiplying AB and CA by C and B respectively)



    For (1), I know (b) and (c) is true, how about (a)? And I dunno how to start (2)
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  2. #2
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    Quote Originally Posted by 450081592 View Post
    (1) if A and B are nonsingular matrics, which, if any of the following must be nonsingular?

    (a) A+B (b) (AB)^2 (c) kA (k subeset R)

    justify your answer either be obtaining the inverse or by a counter example to show it need not exist.


    (2) if A,B and C are square matrices, such that AB = In and CA = In, prove that A is nonsigular and B = C = A^-1.
    (Hint: try multiplying AB and CA by C and B respectively)



    For (1), I know (b) and (c) is true,

    (c) is true iff 0\neq k\in \mathbb{R}


    how about (a)?

    What about A = I and B = -I?


    And I dunno how to start (2)
    The hint is a very good place where to start: AB=I=CA\Longrightarrow C=CI=C(AB)=(CA)B=...

    Tonio
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