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Math Help - Diagonal matrix - general question

  1. #1
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    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?

    <br />
\begin{bmatrix}<br />
a & 0 & 0 \\ <br />
0 & b & 0 \\ <br />
0 & 0 & c<br />
\end{bmatrix}<br />

    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...

    <br />
\begin{bmatrix}<br />
 \frac{1}{a} & 0 & 0 \\ <br />
 0 & \frac{1}{b} & 0 \\ <br />
 0 & 0 & \frac{1}{c}<br />
\end{bmatrix}<br />

    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!
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  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
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    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?
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