Matrices 📊

Numbers ko grid mein arrange karo aur powerful operations karo — real world problems ka ek amazing tool!

🏪 Ek Dukaan ki Inventory

Socho ek dukaan teen items (Apple, Banana, Mango) sell karti hai aur do branches hain. Branch 1 mein 50 apples, 30 bananas, 20 mangoes hain. Branch 2 mein 40 apples, 25 bananas, 35 mangoes hain. Ab agar yeh data ek table (grid) mein likhein toh yeh ek matrix ban jaata hai! Matrix ek systematic way hai data ko organize karne ka — aur phir us data par operations karna mathematical ho jaata hai.

1. Matrix Kya Hai?

Matrix ek rectangular array of numbers hota hai jo rows aur columns mein arranged hota hai. Ek m×n matrix mein m rows aur n columns hote hain.

General notation: A = [aᵢⱼ]ₘₓₙ jahan aᵢⱼ = element at i-th row, j-th column.

2×3 Matrix A [ ] 1 2 3 4 5 6 Col 1 Col 2 Col 3 Row 1 Row 2 Order = 2×3 (2 rows, 3 columns)

2. Types of Matrices

Row Matrix

Sirf ek row: [1, 2, 3] — order 1×n

Column Matrix

Sirf ek column: order m×1

Square Matrix

Rows = Columns: order n×n. Diagonal elements important hote hain!

Zero/Null Matrix

Sab elements zero: O. Identity matrix ka additive inverse.

Identity Matrix (I)

Square matrix jisme diagonal=1, rest=0. A×I = I×A = A.

Diagonal Matrix

Non-diagonal elements sab zero. Sirf diagonal par values.

Symmetric Matrix

Aᵀ = A. Elements mirror hote hain diagonal ke across.

Skew-Symmetric

Aᵀ = -A. Diagonal elements always zero hote hain!

3. Matrix Operations

Addition: Same order ki do matrices add karte hain — element by element. C = A+B means cᵢⱼ = aᵢⱼ + bᵢⱼ

Scalar Multiplication: kA means har element ko k se multiply karo.

Matrix Multiplication: A(m×n) × B(n×p) = C(m×p). Column of B dot product se row of A.

Matrix Multiplication Rule: A = [aᵢⱼ] (m×n), B = [bⱼₖ] (n×p) C = AB = [cᵢₖ] (m×p) cᵢₖ = Σⱼ aᵢⱼ × bⱼₖ Key: A's columns = B's rows (n same hona chahiye!) AB ≠ BA (not commutative in general) A(BC) = (AB)C (associative ✓) (AB)ᵀ = BᵀAᵀ (reverse order!)

✅ Matrix Multiplication Example

A = [[1,2],[3,4]], B = [[5,6],[7,8]]

AB[0][0] = 1×5 + 2×7 = 5+14 = 19

AB[0][1] = 1×6 + 2×8 = 6+16 = 22

AB[1][0] = 3×5 + 4×7 = 15+28 = 43

AB[1][1] = 3×6 + 4×8 = 18+32 = 50

AB = [[19,22],[43,50]]

4. Transpose aur Special Properties

Transpose Aᵀ: rows ko columns bana do aur columns ko rows. (Aᵀ)ᵢⱼ = Aⱼᵢ

/* Important Transpose Properties */ (Aᵀ)ᵀ = A (A+B)ᵀ = Aᵀ + Bᵀ (kA)ᵀ = k(Aᵀ) (AB)ᵀ = BᵀAᵀ ← REVERSE ORDER! /* Every square matrix = Symmetric + Skew-Symmetric */ A = ½(A + Aᵀ) + ½(A - Aᵀ) [Symmetric] [Skew-sym]

5. Invertible Matrices

Ek square matrix A invertible (non-singular) hai agar ∃ matrix B s.t. AB = BA = I. Tab B = A⁻¹.

Condition: |A| ≠ 0 (determinant zero nahi hona chahiye).

6. Practice Questions

Q1. A = [[1,2],[3,4]], B = [[0,1],[1,0]]. Find AB aur BA. Kya AB=BA?

Solution

AB = [[2,1],[4,3]], BA = [[3,4],[1,2]]. AB ≠ BA — matrix multiplication commutative nahi hai!

Q2. Show that A = [[1,2],[2,1]] symmetric hai.

Solution

Aᵀ = [[1,2],[2,1]] = A. ∴ Symmetric ✓

Q3. Express [[2,3],[1,4]] as sum of symmetric and skew-symmetric matrix.

Solution

A = [[2,3],[1,4]], Aᵀ = [[2,1],[3,4]]. Sym = ½(A+Aᵀ) = [[2,2],[2,4]]. Skew = ½(A-Aᵀ) = [[0,1],[-1,0]].