Matrices & Determinants — Applied Maths 12

1. Matrices — Parichay

Matrix ek rectangular arrangement hai numbers ya elements ka, rows aur columns mein. Isko bold capital letter se denote karte hain jaise A, B, C.

A = [aᵢⱼ] jahan i = row number, j = column number
Order: m × n matlab m rows aur n columns

Matrix ke Types

Row Matrix (1×n)

[2 5 -1 3]

Sirf ek row hoti hai

Column Matrix (m×1)

⎡4⎤
⎢7⎥
⎣2⎦

Sirf ek column hota hai

Square Matrix (n×n)

⎡1 2⎤
⎣3 4⎦

Rows = Columns

Identity Matrix (I)

⎡1 0⎤
⎣0 1⎦

Diagonal = 1, baki = 0

Zero/Null Matrix (O)

⎡0 0⎤
⎣0 0⎦

Sare elements 0

Diagonal Matrix

⎡3 0 0⎤
⎢0 5 0⎥
⎣0 0 7⎦

Off-diagonal = 0

2. Matrix Operations

2.1 Addition aur Subtraction

A + B = [aᵢⱼ + bᵢⱼ] (same order honi chahiye dono ki)
A − B = [aᵢⱼ − bᵢⱼ]

Udaharan: Addition

A = [[1,2],[3,4]] aur B = [[5,6],[7,8]]
A + B = [[1+5, 2+6],[3+7, 4+8]] = [[6,8],[10,12]]

2.2 Scalar Multiplication

kA = [k·aᵢⱼ] — har element ko k se multiply karo

Udaharan: Scalar Multiplication

3 × [[1,-2],[0,4]] = [[3,-6],[0,12]]

2.3 Matrix Multiplication

A (m×n) aur B (n×p) ko multiply kar sakte hain. Result (m×p) hoga. Ye tabhi possible hai jab A ki columns = B ki rows.

Cᵢⱼ = Σₖ aᵢₖ · bₖⱼ (i-th row × j-th column, element-wise multiply karke add karo)

Udaharan: 2×2 Multiplication

A = [[1,2],[3,4]] × B = [[5,6],[7,8]]
C₁₁ = 1×5 + 2×7 = 5+14 = 19
C₁₂ = 1×6 + 2×8 = 6+16 = 22
C₂₁ = 3×5 + 4×7 = 15+28 = 43
C₂₂ = 3×6 + 4×8 = 18+32 = 50
A×B = [[19,22],[43,50]]

⚠️ Important: Matrix multiplication commutative NAHI hota! AB ≠ BA generally

2.4 Transpose of a Matrix

Aᵀ = [aⱼᵢ] — rows ko columns aur columns ko rows banao
If A = [[1,2,3],[4,5,6]] then Aᵀ = [[1,4],[2,5],[3,6]]

3. Determinants

Sirf square matrices ka determinant hota hai. Isko |A| ya det(A) likhte hain.

3.1 2×2 Determinant

|A| = |a b| = ad − bc
|c d|

Udaharan

A = [[3,1],[5,2]]
|A| = 3×2 − 1×5 = 6 − 5 = 1

3.2 3×3 Determinant (Expansion along row 1)

|A| = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)

Udaharan: 3×3 Determinant

A = [[1,2,3],[0,1,4],[5,6,0]]
|A| = 1(1·0 − 4·6) − 2(0·0 − 4·5) + 3(0·6 − 1·5)
= 1(0−24) − 2(0−20) + 3(0−5)
= −24 + 40 − 15 = 1

3.3 Properties of Determinants

Property	Statement
Transpose	\|Aᵀ\| = \|A\|
Row swap	Do rows swap karo → sign change hoga
Proportional rows	Koi 2 rows proportional hain → \|A\| = 0
Scalar	\|kA\| = kⁿ\|A\| for n×n matrix
Product	\|AB\| = \|A\|·\|B\|
Singular	\|A\| = 0 → matrix singular (non-invertible)

3.4 Inverse of a Matrix

A⁻¹ = adj(A) / |A| (tabhi exist karta hai jab |A| ≠ 0)

2×2 case: If A = [[a,b],[c,d]] then A⁻¹ = (1/|A|)[[d,-b],[-c,a]]

Udaharan: Inverse

A = [[4,7],[2,6]]
|A| = 4×6 − 7×2 = 24 − 14 = 10
A⁻¹ = (1/10)[[6,−7],[−2,4]] = [[0.6,−0.7],[−0.2,0.4]]
Check: A · A⁻¹ = I ✓

4. System of Linear Equations

n equations aur n unknowns wala system. Matrix form: AX = B

If |A| ≠ 0: X = A⁻¹B (unique solution — consistent) If |A| = 0: ya infinite solutions ya no solution (check augmented matrix)

4.1 Cramer's Rule

2 ya 3 equations ke liye determinant method se direct solution:

For AX = B:
x = Dx/D, y = Dy/D, z = Dz/D

D = |A|
Dx = D with column 1 replaced by B
Dy = D with column 2 replaced by B
Dz = D with column 3 replaced by B

Udaharan: Cramer's Rule (2 variables)

2x + 3y = 8
x − y = 1

D = |2,3; 1,−1| = 2(−1)−3(1) = −2−3 = −5
Dx = |8,3; 1,−1| = 8(−1)−3(1) = −8−3 = −11
Dy = |2,8; 1,1| = 2(1)−8(1) = 2−8 = −6
x = Dx/D = −11/−5 = 2.2
y = Dy/D = −6/−5 = 1.2
Check: 2(2.2)+3(1.2) = 4.4+3.6 = 8 ✓

5. Boolean Algebra

George Boole ne develop ki — digital circuits aur logical reasoning ke liye. Sirf 2 values: 0 (False) aur 1 (True).

5.1 Basic Operations

Operation	Symbol	Name	Meaning
AND	A · B ya A ∧ B	Conjunction	Dono true tabhi result true
OR	A + B ya A ∨ B	Disjunction	Koi ek true tabhi result true
NOT	A' ya Ā ya ¬A	Complement/Negation	True ↔ False

5.2 Truth Tables

A	B	A AND B	A OR B	A'	A XOR B
0	0	0	0	1	0
0	1	0	1	1	1
1	0	0	1	0	1
1	1	1	1	0	0

5.3 Boolean Laws

Law	AND form	OR form
Identity	A · 1 = A	A + 0 = A
Null/Domination	A · 0 = 0	A + 1 = 1
Idempotent	A · A = A	A + A = A
Complement	A · A' = 0	A + A' = 1
Double negation	(A')' = A
Commutative	AB = BA	A+B = B+A
Associative	(AB)C = A(BC)	(A+B)+C = A+(B+C)
Distributive	A(B+C) = AB+AC	A+BC = (A+B)(A+C)
Absorption	A(A+B) = A	A+AB = A

5.4 De Morgan's Theorems

(A + B)' = A' · B' → OR ka complement = AND of complements (A · B)' = A' + B' → AND ka complement = OR of complements

Verification

A=1, B=0:
(A+B)' = (1+0)' = 1' = 0
A'·B' = 0·1 = 0 ✓

(A·B)' = (1·0)' = 0' = 1
A'+B' = 0+1 = 1 ✓

5.5 Simplification — Udaharan

Simplify: F = A'B + AB' + AB

F = A'B + AB' + AB
= A'B + A(B' + B) [AB'+AB = A(B'+B)]
= A'B + A·1 [B'+B=1]
= A'B + A
= (A'+A)(A+B) [Distributive]
= 1·(A+B)
= A + B

📝 Important Formulas Summary

|2×2 Matrix| = ad − bc
A⁻¹ = adj(A)/|A| (only if |A|≠0)
AX = B → X = A⁻¹B (matrix method)
Cramer's: x=Dx/D, y=Dy/D
De Morgan: (A+B)' = A'B', (AB)' = A'+B'

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Matrices & Determinants 📐

1. Matrices — Parichay

Matrix ke Types

Row Matrix (1×n)

Column Matrix (m×1)

Square Matrix (n×n)

Identity Matrix (I)

Zero/Null Matrix (O)

Diagonal Matrix

2. Matrix Operations

2.1 Addition aur Subtraction

Udaharan: Addition

2.2 Scalar Multiplication

Udaharan: Scalar Multiplication

2.3 Matrix Multiplication

Udaharan: 2×2 Multiplication

2.4 Transpose of a Matrix

3. Determinants

3.1 2×2 Determinant

Udaharan

3.2 3×3 Determinant (Expansion along row 1)

Udaharan: 3×3 Determinant

3.3 Properties of Determinants

3.4 Inverse of a Matrix

Udaharan: Inverse

4. System of Linear Equations

4.1 Cramer's Rule

Udaharan: Cramer's Rule (2 variables)

5. Boolean Algebra

5.1 Basic Operations

5.2 Truth Tables

5.3 Boolean Laws

5.4 De Morgan's Theorems

Verification

5.5 Simplification — Udaharan

Simplify: F = A'B + AB' + AB

📝 Important Formulas Summary