Diagonal, Jordan Normal and And Inverses

[heshbon]

1) Is a diagonal matrix always invertable?
2) Is an Invertable matrix always Diagonalizable?
3) Is a matrix in jordan normal form always invertable

The answers are prob straight foward but I am confused.

 

[heshbon]

Jordan Normal, Diagonal and Inverses

Homework Statement

1) Is a diagonal matrix always invertable?
2) Is an Invertable matrix always Diagonalizable?
3) Is a matrix in jordan normal form always invertable

The answers are prob straight foward but I am confused.

Homework Equations

The Attempt at a Solution

 

[heshbon]

I've had a thought about this, I guess what I am looking for is some sort of relation between the characteristic polinomial and the determinant of a the matrix.

 

[Random Variable]

1) No. Let $$d_{i}$$ be the diagonal entries of a diagonal matrix The inverse (if it exists) consists of the values $$\frac {1}{d_{i}}$$ along the diagonal. Therefore, all $$d_{i}$$'s must be nonzero for the inverse to exist.

2) No. If an nxn matrix is diagonalizable, it must have n linearly independent eigenvectors. $$A = \left(\begin{array}{cc}1&0\\1&1\end{array}\right)$$ is invertibile but not diagonalizable.

3) I don't know.

 

[HallsofIvy]

Is
$$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 0\end{bmatrix}$$
invertible?

 

[heshbon]

Thanks greatly.

That clears it all up.