# Linear Algebra: A matrix vs a determinant

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:

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. –  Arturo Magidin Jun 9 '11 at 19:51
So I can use bracket ( "[ ]" ) notation too right? –  Virtuoso Jun 9 '11 at 19:53
I think the convention is that $||$ is used for the determinant of the matrix, and $()$ or $[]$ for the matrix itself. –  Joel Cohen Jun 9 '11 at 19:53
@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. –  Arturo Magidin Jun 9 '11 at 19:54
@Arturo: do you mind if I ask what the {ccr} means in your code for the matrix (etc) above? –  amWhy Jun 9 '11 at 20:50

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... :D –  Ｊ. Ｍ. Nov 26 '11 at 16:50
Then we take "||" in latex instead of "\|". –  Phira Nov 26 '11 at 17:02