Thread: linear algebra: invertible matrices, eigenvalues, diagonalizable

1. linear algebra: invertible matrices, eigenvalues, diagonalizable

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

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

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

I have no idea where to begin for question 2 & 3.
Please give me some hints for question 2 and question 3.

2. Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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

3. Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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

4. Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

Originally Posted by deniselim17
Question 3:
Let $A, B$ be $n$ by $n$ real matrices such that $AB=BA$.
If $A$ is diagonalizable, show that $B$ 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.

5. Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

Originally Posted by 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 " $A$ has two distinct eigenvalues", will it be the same?

6. Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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

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

7. Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

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