4×4 matrix determinant (first‑row expansion, minors inline)

Given 4×4 matrix \[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} \] first‑row Laplace expansion gives the determinant of matrix \( A \) as: \[ \det(A) = a_{11}\det(M_{11}) - a_{12}\det(M_{12}) + a_{13}\det(M_{13}) - a_{14}\det(M_{14}) \] where minors \(M_{ij}\) (delete row i, column j): \[ M_{11}= \begin{bmatrix} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{bmatrix}\] \[ M_{12}= \begin{bmatrix} a_{21} & a_{23} & a_{24} \\ a_{31} & a_{33} & a_{34} \\ a_{41} & a_{43} & a_{44} \end{bmatrix}\] \[ M_{13}= \begin{bmatrix} a_{21} & a_{22} & a_{24} \\ a_{31} & a_{32} & a_{34} \\ a_{41} & a_{42} & a_{44} \end{bmatrix}\] \[ M_{14}= \begin{bmatrix} a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ a_{41} & a_{42} & a_{43} \end{bmatrix}\]

\(\det \begin{bmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{bmatrix}\)

first row: a·det(M₁₁) − b·det(M₁₂) + c·det(M₁₃) − d·det(M₁₄)

\(\text{entries } \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix}\)
row 1 (a₁₁ … a₁₄)
row 2 (a₂₁ … a₂₄)
row 3 (a₃₁ … a₃₄)
row 4 (a₄₁ … a₄₄)
\(\text{precision}\)
\(\det(A) = a_{11}\det(M_{11}) - a_{12}\det(M_{12}) + a_{13}\det(M_{13}) - a_{14}\det(M_{14})\)

computed via first‑row expansion (3×3 minors).

useful links