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So I'm reading through my linear algebra textbook to review for my final, and happened upon this statement:

The determinant of a matrix with positive entries must be positive.

Off the top of my head, I can think of an exception to this:

$$A=\begin{bmatrix}1 & 2\\8 & 3\end{bmatrix}$$

where $\det A= (1\cdot 3) - (2\cdot 8) = 3 - 16 = -13$

Am I misinterpreting what I am reading, or is this a misprint?

The book is Elementary Linear Algebra, 2nd ed. by Spence, Insel, Friedberg

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I suspect the adjective "triangular' was missing from the statement. –  David Mitra Dec 15 '12 at 23:56
You're not misinterpreting - it's nonsense. –  Billy Dec 16 '12 at 0:02
Can you point the page? –  Artem Dec 16 '12 at 0:16
Perhaps it meant to say "eigenvalues" instead of "entries?" –  Ryan Dec 16 '12 at 7:08

4 Answers 4

up vote 7 down vote accepted

The sentence, applied to square matrices - in general - is wrong: This must be a misprint, unless the sentence appears in a specific context with other qualifications you may have left out.

For example, if the discussion is about square triangular matrices, whose non-zero entries are all positive (or with only positive entries on the diagonal), then the statement is true.

So, since I haven't the text to refer to, to examine at what point in the text, and in what context, the sentence appears, I cannot say for sure.

But if it is a global statement about the determinant of all square matrices with all positive entries, then the sentence is blatantly not true.

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I can't imagine a triangular matrix with all positive entries, except in the $1 \times 1$ case. –  Erick Wong Dec 16 '12 at 1:42
@Erick: I meant a triangular matrix whose non-zero entries are all positive. –  amWhy Dec 16 '12 at 1:45

I think the book has a misprint because the sentence, applied to square matrices in general is false.

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I have tried to improve the readability of your answer, but even though, it adds nothing to the other answers. You should try to improve the answer to make it unique or think of deleting it. –  robjohn Dec 16 '12 at 23:49

My guess is that the book was referring to positive-definite matrices, which are often just called "positive matrices". This is not the same as having all positive entries. The simplest definition of positive-definite for matrices is that all the eigenvalues are positive. In this case, it is clear that the determinant is also positive, since the determinant of a matrix is the product of its eigenvalues (counted with algebraic multiplicities).

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The book is wrong.

Take any square matrix with size $\ge2$ and all entries positive. If its determinant is positive, interchange any two rows. Then the determinant will become negative. So, it is always possible to obtain a negative-determinant matrix with positive entries from a positive-determinant one with positive entries.

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