# Diagonal matrix - general question

• Feb 6th 2011, 10:00 AM
Diagonal matrix - general question
So this is a general matrix question, we are working on inverses. The question ask, for which values of a, b, and c is this matrix invertible?

$
\begin{bmatrix}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c
\end{bmatrix}
$

I said, this matrix is invertible when a = 1, b = 1, c = 1, that would mean this matrix has a rref and thus, it has an inverse. Also, the determinant of this matrix, I believe, is simply: abc. That would mean that a, b and c cannot = 0 by any means, if we want this to be invertible. Is this correct?

if thats the case then the inverse matrix would be...

$
\begin{bmatrix}
\frac{1}{a} & 0 & 0 \\
0 & \frac{1}{b} & 0 \\
0 & 0 & \frac{1}{c}
\end{bmatrix}
$

Is this correct?

Then it asks an even more general question: For which values of the diagonal elements is a diagonal matrix, of arbitrary size, invertible?

Help on this would be much appreciated!
• Feb 6th 2011, 10:42 AM
Tinyboss
You are correct that the determinant of a diagonal matrix is the product of its diagonal entries. So, what condition on a set of numbers $\{a_1,a_2,\dots,a_n\}$ will ensure that their product is nonzero?