📘 Class 12 – Matrices (Advanced Notes)

Learn every concept of Matrices with diagrams, formulas, and examples — made for lifetime understanding!

🔹 What is a Matrix?

A Matrix is a rectangular arrangement of numbers (or functions) in rows and columns. It’s widely used in linear algebra, physics, statistics, and data science.

a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃

Representation of a 3×3 Matrix.

🔹 Types of Matrices

🔹 Matrix Operations & Formulas

Addition: A + B = [aᵢⱼ + bᵢⱼ]
Scalar Multiplication: kA = [k × aᵢⱼ]
Multiplication: (AB)ᵢⱼ = Σ aᵢₖ × bₖⱼ
Transpose: (Aᵀ)ᵢⱼ = Aⱼᵢ
Determinant (2×2): |A| = ad − bc
Determinant (3×3): Expand along row or column.

🔹 Theorems on Matrices

Theorem 1: If A and B are square matrices of same order, then |AB| = |A| × |B|.

Theorem 2: (Aᵀ)⁻¹ = (A⁻¹)ᵀ

🧮 Solved Examples (Step-by-Step)

Example 1: Find determinant of A = [[2,3],[1,4]]

Solution: |A| = (2×4) − (3×1) = 8 − 3 = 5

Example 2: If A = [[1,2],[3,4]], find A⁻¹.

Solution: |A| = −2, adj(A) = [[4, −2],[−3, 1]], A⁻¹ = (1/|A|) adj(A) = (−½)×adj(A)

Example 3: Verify (AB)ᵀ = BᵀAᵀ.

Take A=[[1,2],[3,4]], B=[[2,0],[1,2]] → LHS=(AB)ᵀ=[[4,4],[10,8]]ᵀ=[[4,10],[4,8]], RHS=BᵀAᵀ=[[4,10],[4,8]] ✅

📝 Summary Table

ConceptFormula / Property
Transpose(Aᵀ)ᵀ = A
Determinant|AB| = |A||B|
InverseA⁻¹ = (1/|A|) adj(A)
IdentityAI = IA = A

📘 Practice Questions

  1. Find determinant of A = [[3,2],[1,4]].
  2. If A = [[2,0],[1,3]], find A⁻¹.
  3. Show that (AB)ᵀ = BᵀAᵀ.
  4. Find X: 2X + [[1,3],[2,4]] = [[5,7],[8,10]]

Answer Key