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.

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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**deniselim17** 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 $\displaystyle \bigl(A^{-1} + B^{-1}\bigr)^{-1} = B(A+B)^{-1}A.$

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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**deniselim17** 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 $\displaystyle A=\begin{bmatrix}{\;\;0}&{1}\\{-1}&{0}\end{bmatrix}\;,\; B=\begin{bmatrix}{0}&{0}\\{0}&{0}\end{bmatrix}$

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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**deniselim17** 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.

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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**Opalg** 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?

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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**FernandoRevilla** 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??

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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**deniselim17** If I add another condition for $A$ and $B$ are nonzero matrices??

Quote:

Originally Posted by

**deniselim17** 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.