My text book is called "Linear Algebra and its applications" by David C. Lay.
I am just wondering why the textbook uses the absolute value symbol when it wants us to compute determinants. For example, for some matrix A, the determinant is represented as |A| (Chapter 3.1 Determinants).
And yet 2 sections later (3.3 Volume), we actually use absolute values around determinants, which is represented like this: | det(A) |. So if we wanted to compute the determinant of Matrix A as well as have the absolute value of it, it would look like ||A||?
I just don't see why the text book would choose to denote determinants using the same symbols as absolute values. Why would they not choose a different representation of determinants?
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$\begingroup$The notation $|A|$ is used to denote the determinant of the matrix $A$. This is especially useful if you are currently calculating the determinant, for example $$\begin{vmatrix}3 & 4 \\ 2 & 1\end{vmatrix} = \det\begin{pmatrix}3 & 4 \\ 2 & 1\end{pmatrix}$$ The notation $\|A\|$ is typically the norm of the matrix for whatever norm you are using. On the otherhand, $\det(A)$ is another notation for the determinant of the matrix $A$. If you want the absolute value of the determinant, I would use the same notation as your textbook and use $|\det(A)|$. I would not suggest using $\|A\|$ do denote the absolute value of the determinant since there's a confusion with the norm.
To be fair to the textbook, there really isn't a chance of confusion. There is no such thing as the "absolute value" of a matrix. $|A|$ always means determinant.
$\endgroup$ 2 $\begingroup$It's simple convenience. The determinant of a matrix is at least a bit like an absolute value, since matrices that expand volumes have determinants greater than $1$ and vice versa. But it's not properly an absolute value, norm for those who know the word, since lots of nonzero matrices have zero determinant.
To further complicate things, in analysis courses you'll begin to encounter proper absolute values on matrices, the simplest of which is just the matrix's length as a vector in $\mathbb{R}^{n^2}$. Another is the operator norm, which is most often denoted exactly by $||A||$. For that reason and for clarity more generally you should never use, nor see, $||A||$ to mean $|\det A|$.
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