Matrices and Determinants

2 × 2 matrices

Determinant

det [\begin{matrix} a & b \\ c & d \end{matrix}] = a d - b c

Inverse

A = [\begin{matrix} a & b \\ c & d \end{matrix}] \Rightarrow A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]

ad − bc ≠ 0

Determinant

det [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] = a (e i - f h) - b (d i - f g) + c (d h - e g)

Solution

x = \frac{D_{x}}{D} y = \frac{D_{y}}{D}

D = a₁b₂ − a₂b₁, Dₓ = c₁b₂ − c₂b₁, Dᵧ = a₁c₂ − a₂c₁

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