Definition:Determinant/Matrix/Order 3

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Definition

Let $\mathbf A = \left[{a}\right]_3$ be a square matrix of order $3$.

That is, let:

$\mathbf A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$


The determinant of $\mathbf A$ is given by:

$\det \left({\mathbf A}\right) = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$


Then:

\(\displaystyle \det \left({\mathbf A}\right) = \ \ \) \(\displaystyle \operatorname{sgn} \left({1, 2, 3}\right) a_{1 1} a_{2 2} a_{3 3}\) \(+\) \(\displaystyle \operatorname{sgn} \left({1, 3, 2}\right) a_{1 1} a_{2 3} a_{3 2}\)
\(\displaystyle \) \(+\) \(\displaystyle \operatorname{sgn} \left({2, 1, 3}\right) a_{1 2} a_{2 1} a_{3 3}\)
\(\displaystyle \) \(+\) \(\displaystyle \operatorname{sgn} \left({2, 3, 1}\right) a_{1 2} a_{2 3} a_{3 1}\)
\(\displaystyle \) \(+\) \(\displaystyle \operatorname{sgn} \left({3, 1, 2}\right) a_{1 3} a_{2 1} a_{3 2}\)
\(\displaystyle \) \(+\) \(\displaystyle \operatorname{sgn} \left({3, 2, 1}\right) a_{1 3} a_{2 2} a_{3 1}\)
\(\displaystyle = \ \ \) \(\displaystyle a_{1 1} a_{2 2} a_{3 3}\) \(-\) \(\displaystyle a_{1 1} a_{2 3} a_{3 2}\)
\(\displaystyle \) \(-\) \(\displaystyle a_{1 2} a_{2 1} a_{3 3}\)
\(\displaystyle \) \(+\) \(\displaystyle a_{1 2} a_{2 3} a_{3 1}\)
\(\displaystyle \) \(+\) \(\displaystyle a_{1 3} a_{2 1} a_{3 2}\)
\(\displaystyle \) \(-\) \(\displaystyle a_{1 3} a_{2 2} a_{3 1}\)

and thence in a single expression as:

$\displaystyle \det \left({\mathbf A}\right) = \frac 1 6 \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^3 \sum_{k \mathop = 1}^3 \sum_{r \mathop = 1}^3 \sum_{s \mathop = 1}^3 \sum_{t \mathop = 1}^3 \operatorname{sgn} \left({i, j, k}\right) \operatorname{sgn} \left({r, s, t}\right) a_{i r} a_{j s} a_{k t}$

where $\operatorname{sgn} \left({i, j, k}\right)$ is the sign of the permutation $\left({i, j, k}\right)$ of the set $\left\{{1, 2, 3}\right\}$.


The values of the various instances of $\operatorname{sgn} \left({\lambda_1, \lambda_2, \lambda_3}\right)$ are obtained by applications of Parity of K-Cycle.


Using Summation Convention

The determinant of a square matrix of order $3$ $\mathbf A$ can be expressed using the summation convention as:

$\det \left({\mathbf A}\right) = \dfrac 1 6 \operatorname{sgn} \left({i, j, k}\right) \operatorname{sgn} \left({r, s, t}\right) a_{i r} a_{j s} a_{k t}$


Note that there are six indices which appear twice, and so six summations are assumed.


Sources