Nilpotent Matrices: Invertibility and Transpose Proof

[tom_jerry122]

Homework Statement

A. Are nilpotent matrices invertible ? Prove your answer.

B. If A is nilpotent, what can you say about (A)^τ ? Prove your answer.

C. If A is nilpotent, show I-A is invertible.

Homework Equations

NONE

The Attempt at a Solution

A. I know invertible matrix are - AB = BA = I

B. I took a nilpotent matrix
A = [ 0 1
0 0 ]
Its transpose is -
(A)^τ = [ 0 0
1 0 ]
And the transpose is still a nilpotent matrix because
(A^τ)^2 = [ 0 0
0 0 ]
But I don't know if its true for all and it says prove your answer.

C. No idea

 

[Oster]

A matrix is invertible if and only if it has non-zero determinant. What can you say about the determinant of a nilpotent matrix?
C. You could prove this by assuming it is false i.e. I-A is not invertible and then proceeding.

 

[tom_jerry122]

A matrix is invertible if and only if it has non-zero determinant. What can you say about the determinant of a nilpotent matrix?
C. You could prove this by assuming it is false i.e. I-A is not invertible and then proceeding.

A nilpotent matrix has determinant 0 since its diagonals are all 0 (Eigen values are 0). So the inverse would be 0 too .