Determinants and their Applications — Formula Sheet

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

\Delta = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})

x = \frac{D_1}{D}, y = \frac{D_2}{D}, z = \frac{D_3}{D}

For system ax+by=e, cx+dy=f: x = D₁/D, y = D₂/D where D₁ replaces x-column with constants, D₂ replaces y-column.

If determinant = 0, the system may not have a unique solution.

Cofactor = (−1)^(i+j) × Minor.