# Thread: Diagonal matrix - general question

1. ## 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?

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

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...

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

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!

2. 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 $\displaystyle \{a_1,a_2,\dots,a_n\}$ will ensure that their product is nonzero?

diagonal, general, matrix, question 