ThmDex – An index of mathematical definitions, results, and conjectures.

Complex arithmetic expression for the determinant of a 3-by-3 complex square matrix

Formulation 0

Let $A \in \mathbb{C}^{3 \times 3}$ be a D6159: Complex square matrix such that

(i)	\begin{equation} A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \end{equation}

Then \begin{equation} \text{Det} A = a e i - a f h + c d h - b d i + b f g - c e g \end{equation}

Proofs

Proof 0

Let $A \in \mathbb{C}^{3 \times 3}$ be a D6159: Complex square matrix such that

(i)	\begin{equation} A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \end{equation}

Follows by computing the definition using the permutation signs from R5522: Signs of length-3 standard permutations. $\square$