True/False on Matrices: Answers & Explanations

[EvLer]

1. If B is formed from A by exchanging two rows then B is similar to A
2. If a triangular matrix is similar to diagonal, it is already diagonal
3. If A and B are diagonalizable so is AB
4. Every invertible matrix can be diagonalized

My answers:
1. F: eigenvalues change, so the Jordan form would be different
2. I want to say true, but not entirely sure: eigenvalues would be the same and the jordan form is same so i guess that the triangular would be diagonal?
3. i would guess true, but not sure how to motivate
4. F, I know that for sure because we had an example in class
[1 -1]
[0 1] which is invertible but not diagonalizable since it has only 1 eigenvector, but how would I motivate it theoretically, i.e. how would I say in general that in certain cases there are not enough eigenvectors even for invertible matrices => not diagonalizable?

Thanks!

 

[mighty2000]

1. I agree with your answer that the statement is false. When two rows are exchanged, the eigenvalues of the matrix may change, and thus the Jordan form would also be different. Therefore, B would not be similar to A.
2. I also agree that the statement is true. Triangular matrices have their eigenvalues on the main diagonal, and similar matrices have the same eigenvalues. Since diagonal matrices have their eigenvalues on the main diagonal as well, a triangular matrix that is similar to diagonal must already be diagonal.
3. The statement is true. If A and B are diagonalizable, then they can be written as A = PDP^-1 and B = QDQ^-1, where D is a diagonal matrix and P and Q are invertible matrices. Then, AB = PDP^-1QDQ^-1 = PDQ^-1, which is also diagonal since D is diagonal. Therefore, AB is also diagonalizable.
4. The statement is false. As you mentioned, there are cases where an invertible matrix cannot be diagonalized, such as when it has repeated eigenvalues. This is because for a matrix to be diagonalizable, it must have a full set of linearly independent eigenvectors, which may not always be the case for an invertible matrix. So, not every invertible matrix can be diagonalized.