ObsidianVault/Wiki/Math/Matricies/Determinant of a Matrix.md

#rs/class/math163 #math

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For a 2 x 2 matrix: $$ \det\begin{pmatrix} a&b\c&d \end{pmatrix} = \begin{vmatrix} a&b\c&d\end{vmatrix} = ad-bc$$ For a 3 x 3 matrix:

\begin{vmatrix} a&b&c\\d&e&f\\g&h&i\end{vmatrix} = a\begin{vmatrix} e&f\\h&i\end{vmatrix} - b\begin{vmatrix} d&f\\g&i\end{vmatrix} + c\begin{vmatrix} d&e\\g&h\end{vmatrix} = a(ei-hf)-b(di-gf)+c(dh-ge)

Properties

$$\begin{vmatrix} a&b\a&b \end{vmatrix} = ab - ba = 0$$ $$\begin{vmatrix} r\cdot a&b\r \cdot c&d \end{vmatrix} = rad - rbc = r(ad - bc) = r \cdot \begin{vmatrix} a&b\c&d \end{vmatrix}$$

$$\begin{vmatrix} 1&0\0&1 \end{vmatrix} = 1$$