# linear algebra: invertible matrices, eigenvalues, diagonalizable

• Jan 8th 2012, 06:10 PM
deniselim17
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.
• Jan 9th 2012, 12:20 AM
Opalg
Re: linear algebra: invertible matrices, eigenvalues, diagonalizable
Quote:

Originally Posted by 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 $\displaystyle \bigl(A^{-1} + B^{-1}\bigr)^{-1} = B(A+B)^{-1}A.$
• Jan 9th 2012, 12:32 AM
FernandoRevilla
Re: linear algebra: invertible matrices, eigenvalues, diagonalizable
Quote:

Originally Posted by 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 $\displaystyle A=\begin{bmatrix}{\;\;0}&{1}\\{-1}&{0}\end{bmatrix}\;,\; B=\begin{bmatrix}{0}&{0}\\{0}&{0}\end{bmatrix}$
• Jan 9th 2012, 01:32 AM
Opalg
Re: linear algebra: invertible matrices, eigenvalues, diagonalizable
Quote:

Originally Posted by 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.
• Jan 9th 2012, 01:37 AM
deniselim17
Re: linear algebra: invertible matrices, eigenvalues, diagonalizable
Quote:

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 "$\displaystyle A$ has two distinct eigenvalues", will it be the same?
• Jan 9th 2012, 06:56 AM
deniselim17
Re: linear algebra: invertible matrices, eigenvalues, diagonalizable
Quote:

Originally Posted by 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??
• Jan 9th 2012, 08:30 AM
FernandoRevilla
Re: linear algebra: invertible matrices, eigenvalues, diagonalizable
Quote:

Originally Posted by 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.