linear algebra: invertible matrices, eigenvalues, diagonalizable

Question 1:

Let be by real matrices.

If and are invertible, show that is also invertible.

I tried this question by using elementary matrices. But I still cannot complete the proof.

Question 2:

Let be by real matrices such that .

Suppose is an eigenvalue of or .

Show that is also eigenvalue of .

Question 3:

Let be by real matrices such that .

If is diagonalizable, show that 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

Quote:

Originally Posted by

**deniselim17** Question 1:

Let

be

by

real matrices.

If

and

are invertible, show that

is also invertible.

Take the inverse of both sides to get

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

Quote:

Originally Posted by

**deniselim17** Question 2: Let

be

by

real matrices such that

.

Suppose

is an eigenvalue of

or

. Show that

is also eigenvalue of

.

The statement is false. Choose for example

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

Quote:

Originally Posted by

**deniselim17** Question 3:

Let

be

by

real matrices such that

.

If

is diagonalizable, show that

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

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 " has two distinct eigenvalues", will it be the same?

Re: linear algebra: invertible matrices, eigenvalues, diagonalizable

Quote:

Originally Posted by

**FernandoRevilla** The statement is false. Choose for example

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

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 "

has two distinct eigenvalues", will it be the same?

I see that you only like true statements.