# Thread: Prove that matrix A is singular

1. ## Prove that matrix A is singular

I am not sure where to begin with a question like this:

Question: Let A be an n * n matrix. Prove that A is singular if and only if lambda = 0 is an eigenvalue of A.

Any guidance will be appreciated.

2. Originally Posted by sparky
I am not sure where to begin with a question like this:

Question: Let A be an n * n matrix. Prove that A is singular if and only if lambda = 0 is an eigenvalue of A.

Any guidance will be appreciated.

$Ax=\lambda x$

If you need more guidance,

$Ax-\lambda x=(A-I\lambda)x=0$

You will have to go the other direction after this too.

3. An alternative:

$A\;\textrm{singular}\Leftrightarrow\det A=0 \Leftrightarrow \det (A-0I)=0\Leftrightarrow 0\;\textrm{is\;eigenvalue\;of\;}A$

Fernando Revilla