Determinant

The determinant of an n×n matrix $\left(\vphantom{ a_{ij}}\right.$aij$\left.\vphantom{ a_{ij}}\right)$ is the sum and difference of certain products of the entries. Specifically,

det(aij) = $\displaystyle \sum_{{\sigma }}^{}$$\displaystyle \left(\vphantom{ -1}\right.$ -1$\displaystyle \left.\vphantom{ -1}\right)^{{sgn\left( \sigma \right)
}}_{}$a1σ$\scriptstyle \left(\vphantom{ 1}\right.$1$\scriptstyle \left.\vphantom{ 1}\right)$a2σ$\scriptstyle \left(\vphantom{ 2}\right.$2$\scriptstyle \left.\vphantom{ 2}\right)$ ... anσ$\scriptstyle \left(\vphantom{ n}\right.$n$\scriptstyle \left.\vphantom{ n}\right)$

where σ ranges over all the permutations of $\left\{\vphantom{ 1,2,\ldots
,n}\right.$1, 2,…, n$\left.\vphantom{ 1,2,\ldots
,n}\right\}$ and $\left(\vphantom{ -1}\right.$ -1$\left.\vphantom{ -1}\right)^{{sgn\left( \sigma \right) }}_{}$ = ±1, depending on whether σ is an even or odd permutation.

Note that this operation applies to square matrices only.

$\blacktriangleright$ To compute the determinant of a square matrix

1.
Place the insertion point in the matrix.

2.
From the Matrices submenu, choose Determinant.

$\left[\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right.$$\begin{array}{cc}
a & b \\
c & d
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right]$, determinant: ad - bc


$\left[\vphantom{
\begin{array}{ccc}
a_{1,1} & a_{1,2} & a_{1,3} \\
a_{2,1} & a_{2,2} & a_{2,3} \\
a_{3,1} & a_{3,2} & a_{3,3}
\end{array}
}\right.$$\begin{array}{ccc}
a_{1,1} & a_{1,2} & a_{1,3} \\
a_{2,1} & a_{2,2} & a_{2,3} \\
a_{3,1} & a_{3,2} & a_{3,3}
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
a_{1,1} & a_{1,2} & a_{1,3} \\
a_{2,1} & a_{2,2} & a_{2,3} \\
a_{3,1} & a_{3,2} & a_{3,3}
\end{array}
}\right]$, determinant:
a1, 1a2, 2a3, 3 - a1, 1a2, 3a3, 2 - a2, 1a1, 2a3, 3
     + a2, 1a1, 3a3, 2 + a3, 1a1, 2a2, 3 - a3, 1a1, 3a2, 2


$\left[\vphantom{
\begin{array}{rrr}
-85 & -55 & -37 \\
-35 & 97 & 50 \\
79 & 56 & 49
\end{array}
}\right.$$\begin{array}{rrr}
-85 & -55 & -37 \\
-35 & 97 & 50 \\
79 & 56 & 49
\end{array}$$\left.\vphantom{
\begin{array}{rrr}
-85 & -55 & -37 \\
-35 & 97 & 50 \\
79 & 56 & 49
\end{array}
}\right]$, determinant: -121529

You can also denote the determinant by

det$\displaystyle \left[\vphantom{
\begin{array}{rrr}
-85 & -55 & -37 \\
-35 & 97 & 50 \\
79 & 56 & 49
\end{array}
}\right.$$\displaystyle \begin{array}{rrr}
-85 & -55 & -37 \\
-35 & 97 & 50 \\
79 & 56 & 49
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{rrr}
-85 & -55 & -37 \\
-35 & 97 & 50 \\
79 & 56 & 49
\end{array}
}\right]$

where UserInputdet is typed while in mathematics mode. This string is a multicharacter function name. When typed in mathematics, it turns gray when the t is typed. This function can also be chosen from the list under itbpF0.3009in0.3009in0.0701infunction.wmf.