Determinant

$determinant$

det (A) = σ \in σ^{'} \sum ε (σ) a_{1 i_{1}}, a_{2 i_{2}}, \dots a_{n i_{n}}

det A = a_{11} a_{22} a_{33} + a_{13} a_{21} a_{32} + a_{12} a_{23} a_{31} - a_{13} a_{22} a_{31} - a_{12} a_{21} a_{33} - a_{11} a_{23} a_{32}

det (A^{T}) = det (A)

d e t ⎝ ⎛ a_{11} ⋮ ⋮ a_{n 1} \dots \dots ν \sum c_{ν} a_{1 j, ν} ⋮ ⋮ ν \sum c_{ν} a_{nj, ν} \dots \dots a_{1 n} ⋮ ⋮ a_{nn} ⎠ ⎞

= \sum_{ν} c_{ν} d e t ⎝ ⎛ a_{11} ⋮ ⋮ a_{n 1} \dots \dots a_{1 j, ν} ⋮ ⋮ a_{nj, ν} \dots \dots a_{1 n} ⋮ ⋮ a_{nn} ⎠ ⎞

det ⎝ ⎛ a_{11} ⋮ ⋮ a_{n 1} \dots \dots a_{1 k} ⋮ ⋮ a_{nk} \dots \dots a_{1 j} ⋮ ⋮ a_{nj} \dots \dots a_{1 n} ⋮ ⋮ a_{nn} ⎠ ⎞ = - det ⎝ ⎛ a_{11} ⋮ ⋮ a_{n 1} \dots \dots a_{1 j} ⋮ ⋮ a_{nj} \dots \dots a_{1 k} ⋮ ⋮ a_{nk} \dots \dots a_{1 n} ⋮ ⋮ a_{nn} ⎠ ⎞

det ⎝ ⎛ a_{11} ⋮ ⋮ a_{n 1} \dots \dots 0 ⋮ ⋮ 0 \dots \dots a_{1 n} ⋮ ⋮ a_{nn} ⎠ ⎞ = 0

Determinant of matrix that contains two same columns is 0

det (A B) = det A det B

det (A^{- 1}) = (det A)^{- 1}

Partial matrix by removing

i

,

j

columns:

C_{ij} = ⎝ ⎛ a_{11} ⋮ a_{i - 1, 1} a_{i + 1, 1} ⋮ a_{n 1} \dots \dots \dots \dots a_{1, j - 1} ⋮ a_{i - 1, j - 1} a_{i + 1, j - 1} ⋮ a_{n, j - 1} a_{1, j + 1} ⋮ a_{i - 1, j + 1} a_{i + 1, j + 1} ⋮ a_{n, j + 1} \dots \dots \dots \dots a_{1 n} ⋮ a_{i - 1, n} a_{i + 1, n} ⋮ a_{nn} ⎠ ⎞

Adjugate of

A

:

A_{ij} := (- 1)^{i + j} det C_{ij}

det A = \sum_{i = 1}^{n} a_{ik} A_{ik}

det A = \sum_{i = 1}^{n} a_{l i} A_{l i}

A : m \times n

matrix,

B : n \times m

matrix ,

C = A B

1. m = n \Rightarrow det C = det A det B 2. m > n \Rightarrow det C = 0 3. m < n \Rightarrow d e tC = 1 \leq α_{1} \dots < α_{m} \leq n \sum det ⎝ ⎛ a_{1 α_{1}} a_{1 α_{2}} a_{1 α_{m}} a_{2 α_{1}} a_{2 α_{2}} \dots a_{2 α_{m}} \dots \dots \dots \dots a_{m α_{1}} a_{m α_{2}} a_{m α_{1}} ⎠ ⎞ det ⎝ ⎛ b_{1 α_{1}} b_{1 α_{2}} b_{1 α_{m}} b_{2 α_{1}} b_{2 α_{2}} \dots b_{2 α_{m}} \dots \dots \dots \dots b_{m α_{1}} b_{m α_{2}} b_{m α_{1}} ⎠ ⎞