Revise: Matrices
Matrix notation, addition, subtraction, scalar and matrix multiplication, determinants, and the inverse of a 2×2 matrix for solving simultaneous equations.
Add/subtract matching entries. Requires same order.
[[1,2],[3,4]]+[[5,6],[7,8]]=[[6,8],[10,12]].
(AB)ᵢⱼ = row i of A · column j of B.
[[1,2],[3,4]]×[[5,6],[7,8]]=[[19,22],[43,50]].
A⁻¹ = (1/det A)[[d,−b],[−c,a]].
[[4,7],[2,6]] → A⁻¹=(1/10)[[6,−7],[−2,4]].
Order = rows×columns. aᵢⱼ addresses row i, column j.
2×3 matrix, a₂₃ = last entry of row 2.
Only defined for matrices of the same order.
[[8,3],[2,6]]−[[5,1],[4,2]]=[[3,2],[−2,4]].
Multiply every entry by the scalar.
3×[[2,−1],[3,4]]=[[6,−3],[9,12]].
Swap diagonal, negate off-diagonal, divide by det.
[[2,3],[1,4]] → (1/5)[[4,−3],[−1,2]].
X=A⁻¹B works because A⁻¹A=I.
2x+y=7, x+3y=11 → x=2, y=3.