Romania Clasa a XII-a Mathematics (social sciences)
Chapters: 2
1. Matrices and linear systems
Matrices · Determinants · Linear systems
- Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
- Determinants: Minors, Cofactors, Adjoint, Inverse and Linear Systems – A determinant is one number made from a square matrix, written |A|. For a 2 × 2 matrix it is ad − bc, and it equals the (signed) area made by the columns. For a 3 × 3 matrix we expand along a row using minors and cofactors. |A| = 0 means A is singular and has no inverse. Half of a determinant gives the area of a triangle. The adjoint (transpose of the cofactor matrix) gives A⁻¹ = (adj A)/|A|, and then a system AX = B is solved by X = A⁻¹B. The value of |A| and (adj A)B tell us if a system is consistent.
2. *Algebraic structures
*Groups, rings, fields
- Group Theory: Binary Operations, Groups, Rings and Fields – A group is a set with one operation that is closed, associative, has an identity and gives every element an inverse. Clock arithmetic Z6 shows all of it: tables, subgroups, the order of an element and Lagrange's theorem. Rings and fields add a second operation.