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I would like to clarify something in regards to finding a determinant of a matrix. For example say I have this problem where I have to evaluate the determinant:

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Is the determinant notation simply a reference to the notion that you are supposed to find the determinant?(as in an operation) So in other words, could I write this matrix with bracket notation if I wanted to? (or if there were no instructions, say)

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    Yes: $$\det\left(\begin{array}{ccr}1&7&-3\\1&3&1\\4&8&1\end{array}\right)\quad\text{and}\quad\left|\begin{array}{ccr}1&7&-3\\1&3&1\\4&8&1\end{array}\right|$$mean the same thing; so if the instructions are to "evaluate", then you are to evaluate the determinant of the matrix.2011-06-09
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    So I can use bracket ( "[ ]" ) notation too right?2011-06-09
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    I think the convention is that $||$ is used for the determinant of the matrix, and $()$ or $[]$ for the matrix itself.2011-06-09
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    @Virtuoso: Brackets should *not* be used for determinants, because a lot of books use them for matrices/vectors. Vertical bars are the standard for determinants.2011-06-09
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    @Arturo: do you mind if I ask what the {ccr} means in your code for the matrix (etc) above?2011-06-09
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    @amWhy: When declaring an array, you can then declare how you want the columns to be aligned; I declared an array, then said I wanted the first column `c`entered, the second column `c`entered, and the third column `r`ight justified. The other option is `l`, for left-justified. Note I used `\begin{array}`, rather than some of those "newfangled" `\pmatrix` commands that I never learned when I was learning plain TeX.2011-06-09
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    @Arturo: Thanks! I've seen both (\begin{array} and \pmatrix)...thanks for the clarification! (c: center, etc.)2011-06-09
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    @Arturo: Please explain about plain TeX. Certainly `\begin{array}` is only LaTeX, but `\pmatrix` is plain TeX and always has been. Maybe it's newfangled in LaTeX, but you should have learnt it when learning plain TeX.2011-11-28
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    @MarcvanLeeuwen: I seem to have missed the throwaway comment in *The TeX Book* (bottom of page 176); I always assumed `\pmatrix` had been added in LaTeX.2011-11-28

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Frequent notations for a matrix are

$\begin{pmatrix}a& b\\ c&d\end{pmatrix}$ and $\begin{bmatrix}a& b\\ c&d\end{bmatrix}$.

Frequent notations for determinants are

$\left|\begin{matrix}a& b\\ c&d\end{matrix}\right|$ and $\left\|\begin{matrix}a& b\\ c&d\end{matrix}\right\|$ and, of course, $\det\begin{pmatrix}a& b\\ c&d\end{pmatrix}$ and $\det\begin{bmatrix}a& b\\ c&d\end{bmatrix}$.

Brackets are sometimes used for determinants if the matrix is given by its row or column vectors, so if $v$ and $w$ are vectors with 2 coordinates then $[v,w]$ might be used for the determinant of the corresponding matrix.

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    ...and if we were ever considering the absolute value of a determinant... :D2011-11-26
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    Then we take "||" in latex instead of "\|".2011-11-26