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Math Help - invertible matrices

  1. #1
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    Angry invertible matrices

    I have 2 questions here.

    Let A \in M _{n} ( \mathbb{R}).

    1. If A is an invertible upper triangular matrix, show that A^{-1} also an upper triangular matrix.

    2. If A^{m} =0 for some positive integer m, show that A-I_{n} is invertible and find (A - I_{n})^{-1}.

    I know how to show (1) by using cofactor. Is there any other method without using determinant and adjoin?

    For (2), I dont even know where to start.
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  2. #2
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    Quote Originally Posted by deniselim17 View Post
    I have 2 questions here.

    Let A \in M _{n} ( \mathbb{R}).

    1. If A is an invertible upper triangular matrix, show that A^{-1} also an upper triangular matrix.

    2. If A^{m} =0 for some positive integer m, show that A-I_{n} is invertible and find (A - I_{n})^{-1}.

    I know how to show (1) by using cofactor. Is there any other method without using determinant and adjoin?

    For (2), I dont even know where to start.
    1. For 1\leqslant k\leqslant n, let E_k be the subspace of \mathbb{R}^n spanned by the first k vectors in the standard basis. The condition for A to be upper triangular is that AE_k\subseteq E_k for each k. If A is invertible then its kernel consists only of the zero vector. So AE_k has the same dimension as E_k and therefore AE_k=E_k for each k. Thus E_k = A^{-1}AE_k = A^{-1}E_k, which says that A^{-1} is upper triangular.

    2. If some power of A is 0 then the "binomial series" I_n + A + A^2 + A^3 +\ldots becomes a finite series series which gives you a formula for (A - I_{n})^{-1}.
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