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Thread: linear algebra: invertible matrices, eigenvalues, diagonalizable

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    linear algebra: invertible matrices, eigenvalues, diagonalizable

    Question 1:
    Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices.
    If $\displaystyle A, B$ and $\displaystyle A+B$ are invertible, show that $\displaystyle A^{-1} + B^{-1}$ is also invertible.
    I tried this question by using elementary matrices. But I still cannot complete the proof.

    Question 2:
    Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices such that $\displaystyle AB=0$.
    Suppose $\displaystyle \lambda$ is an eigenvalue of $\displaystyle A$ or $\displaystyle B$.
    Show that $\displaystyle \lambda$ is also eigenvalue of $\displaystyle A+B$.

    Question 3:
    Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices such that $\displaystyle AB=BA$.
    If $\displaystyle A$ is diagonalizable, show that $\displaystyle B$ is diagonalizable.

    I have no idea where to begin for question 2 & 3.
    Please give me some hints for question 2 and question 3.
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    Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

    Quote Originally Posted by deniselim17 View Post
    Question 1:
    Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices.
    If $\displaystyle A, B$ and $\displaystyle A+B$ are invertible, show that $\displaystyle A^{-1} + B^{-1}$ is also invertible.
    $\displaystyle A^{-1} + B^{-1} = A^{-1}(A+B)B^{-1}.$ Take the inverse of both sides to get $\displaystyle \bigl(A^{-1} + B^{-1}\bigr)^{-1} = B(A+B)^{-1}A.$
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    Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

    Quote Originally Posted by deniselim17 View Post
    Question 2: Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices such that $\displaystyle AB=0$.
    Suppose $\displaystyle \lambda$ is an eigenvalue of $\displaystyle A$ or $\displaystyle B$. Show that $\displaystyle \lambda$ is also eigenvalue of $\displaystyle A+B$.
    The statement is false. Choose for example $\displaystyle A=\begin{bmatrix}{\;\;0}&{1}\\{-1}&{0}\end{bmatrix}\;,\; B=\begin{bmatrix}{0}&{0}\\{0}&{0}\end{bmatrix}$
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    Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

    Quote Originally Posted by deniselim17 View Post
    Question 3:
    Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices such that $\displaystyle AB=BA$.
    If $\displaystyle A$ is diagonalizable, show that $\displaystyle B$ is diagonalizable.
    This statement is also false. Take for example A to be the identity nxn matrix and B to be a non-diagonalisable matrix.
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    Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

    Quote Originally Posted by Opalg View Post
    This statement is also false. Take for example A to be the identity nxn matrix and B to be a non-diagonalisable matrix.
    If I change the question to "$\displaystyle A$ has two distinct eigenvalues", will it be the same?
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    Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

    Quote Originally Posted by FernandoRevilla View Post
    The statement is false. Choose for example $\displaystyle A=\begin{bmatrix}{\;\;0}&{1}\\{-1}&{0}\end{bmatrix}\;,\; B=\begin{bmatrix}{0}&{0}\\{0}&{0}\end{bmatrix}$

    If I add another condition for $A$ and $B$ are nonzero matrices??
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    MHF Contributor FernandoRevilla's Avatar
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    Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

    Quote Originally Posted by deniselim17 View Post
    If I add another condition for $A$ and $B$ are nonzero matrices??
    Quote Originally Posted by deniselim17 View Post
    If I change the question to "$\displaystyle A$ has two distinct eigenvalues", will it be the same?
    I see that you only like true statements.
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