Determinants

|A| = ad - bc for matrix

  A =⌈a b⌉ 
     ⌊c d⌋ 2x2 

|A| = a(ei − fh) − b(di − fg) + c(dh − eg) for matrix

  A =⌈a b c⌉ 
     |d e f| 
     ⌊g h i⌋ 3x3

Minor of element a_ij is the determinant of the matrix obtained by deleting the i-th row and j-th column.
Cofactor of element a_ij is (-1)^i+j × Minor of a_ij
⌈ + - + ⌉
| - + - |
⌊ + - + ⌋

|A| = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ (along first row)
|A| = a₁₁C₁₁ + a₂₁C₂₁ + a₃₁C₃₁ (along first column)
|A| = a₁₁M₁₁ - a₁₂M₁₂ + a₁₃M₁₃ (In terms of minors)

1. If any two rows (or columns) of a determinant are identical → |A| = 0.
2. If any two rows (or columns) of a determinant are proportional → |A| = 0.
3. The value of a determinant remains unchanged if its rows and columns are interchanged.
4. If Row/ columns interchange → |A| = -|A|
5. Multiplying a row/column by a scalar k → |A| = k|A|
6. For a matrix A, |kA| = kⁿ|A| where n is the order of the matrix
7. Any row/column can be added to another row/column without changing the value of the determinant.
8. Any row/column zero element → |A| = 0

The adjoint of a matrix A is the transpose of the matrix of cofactors of A.

A⁻¹ = (1/|A|) × adj(A)

Area = (1/2) |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

        1 ⌈x₁ y₁ 1⌉ 
 Area = — |x₂ y₂ 1| 
        2 ⌊x₃ y₃ 1⌋