1.INTRODUCTION TO MATRICES

1.1.Definition

1.A matrix is a rectangular arrangement of numbers in rows and columns.
2.It is denoted by capital letters like A, B, C.
3.If a matrix has m rows and n columns, its order is m×n.
4.Total number of elements = m×n.

Example of 2×3 matrix:

|1|2|3|
|4|5|6|

Order = 2×3

1.2.Elements of a Matrix

1.Element in ith row and jth column is written as aᵢⱼ.
2.For matrix A = [aᵢⱼ], i represents row number and j represents column number.

2.TYPES OF MATRICES

1.Row Matrix → Only one row (1×n)
2.Column Matrix → Only one column (m×1)
3.Rectangular Matrix → m ≠ n
4.Square Matrix → m = n
5.Zero Matrix → All elements are zero
6.Diagonal Matrix → Non-diagonal elements are zero
7.Scalar Matrix → Diagonal elements equal
8.Identity Matrix → Diagonal elements are 1
9.Upper Triangular Matrix → Elements below main diagonal zero
10.Lower Triangular Matrix → Elements above main diagonal zero
11.Symmetric Matrix → Aᵀ = A

Identity Matrix (2×2):

|1|0|
|0|1|

3.ALGEBRA OF MATRICES

3.1.Addition

1.Possible only if matrices have same order.
2.Add corresponding elements.

3.2.Subtraction

1.Orders must be same.
2.Subtract corresponding elements.

3.3.Scalar Multiplication

1.Multiply every element by constant k.
2.kA = [kaᵢⱼ]

3.4.Matrix Multiplication

Condition:
1.Number of columns of first matrix = number of rows of second matrix.
2.If A is m×n and B is n×p → AB is m×p.

Important Properties:

1.AB ≠ BA (Not commutative)
2.A(BC) = (AB)C (Associative)
3.A(B + C) = AB + AC (Distributive)
4.AI = IA = A
5.AO = O

4.TRANSPOSE OF A MATRIX

4.1.Definition

1.Transpose of matrix A is written as Aᵀ.
2.Rows become columns.

4.2.Properties

1.(Aᵀ)ᵀ = A
2.(A + B)ᵀ = Aᵀ + Bᵀ
3.(AB)ᵀ = BᵀAᵀ
4.If A is symmetric → Aᵀ = A

5.APPLICATION:PRICE AND PROFIT MODEL

5.1.Price Matrix

A =
40 50
35 30
400 450

5.2.Profit Row Matrix

B =
4 6 50

5.3.Product BA

Shop A profit = 20370
Shop B profit = 22880

Conclusion: Shop B gives higher total profit.

6.DETERMINANT OF A MATRIX

6.1.Definition

1.Determinant is defined only for square matrices.
2.It is denoted by |A|.

6.2.Determinant of 2×2 Matrix

For

a b
c d

|A| = ad − bc

6.3.Properties

1.|AB| = |A||B|
2.|Aᵀ| = |A|
3.If two rows are equal → determinant = 0
4.If any row is zero → determinant = 0
5.Interchanging two rows changes sign
6.If |A| = 0 → Singular
7.If |A| ≠ 0 → Non-Singular

6.4.Geometrical Meaning

1.Determinant of 2×2 matrix represents area of parallelogram.
2.If determinant = 0 → area = 0 → rows are proportional.

7.INVERSE OF A MATRIX

7.1.Definition

Inverse of A is A⁻¹ such that:

A × A⁻¹ = I

7.2.Condition

Inverse exists only if |A| ≠ 0.

7.3.Formula (2×2)

A⁻¹ = 1/(ad − bc) ×

d −b
−c a

7.4.Properties of Inverse

1.(A⁻¹)⁻¹ = A
2.(AB)⁻¹ = B⁻¹A⁻¹
3.I⁻¹ = I
4.|A⁻¹| = 1/|A|

8.SOLUTION OF LINEAR EQUATIONS

8.1.Matrix Method

AX = B

X = A⁻¹B

8.2.Types of Solutions

1.Unique solution → |A| ≠ 0
2.No solution or infinite solution → |A| = 0

9.CRAMER’S RULE

For

a₁x + b₁y = c₁
a₂x + b₂y = c₂

1.D = a₁b₂ − a₂b₁
2.Dx = c₁b₂ − c₂b₁
3.Dy = a₁c₂ − a₂c₁
4.x = Dx/D
5.y = Dy/D

Condition: D ≠ 0

10.COMMON MISTAKES IN EXAMS

1.Forgetting order condition in multiplication
2.Not checking determinant before inverse
3.Sign mistake in ad − bc
4.Multiplying matrices in wrong order
5.Forgetting to divide by determinant

11.IMPORTANT QUESTIONS (7 QUESTIONS WITH SOLUTIONS)

Q1.Find determinant

|4 3|
|2 5|

Solution:

1.(4×5) = 20
2.(3×2) = 6
3.|A| = 20 − 6 = 14

Answer:14

Q2.Find inverse

|3 1|
|2 1|

Solution:

1.|A| = (3×1 − 1×2)
= 3 − 2
= 1

2.A⁻¹ =

1 −1
−2 3

Answer:

|1 −1|
|−2 3|

Q3.Solve using Matrix Method

2x + 3y = 11
x + y = 5

Solution:

1.From second equation: x = 5 − y
2.Substitute in first
3.2(5 − y) + 3y = 11
4.10 − 2y + 3y = 11
5.10 + y = 11
6.y = 1
7.x = 4

Answer: x = 4,y = 1

Q4.Solve using Cramer’s Rule

x + 2y = 7
3x + y = 11

1.D = (1×1 − 3×2)
= 1 − 6
= −5

2.Dx = (7×1 − 11×2)
= 7 − 22
= −15

3.Dy = (1×11 − 3×7)
= 11 − 21
= −10

4.x = (−15)/(−5) = 3
5.y = (−10)/(−5) = 2

Answer:x = 3,y = 2

Q5.Verify |Aᵀ| = |A|

|2 5|
|3 4|

|A| = (2×4 − 5×3)
= 8 − 15
= −7

Transpose:

|2 3|
|5 4|

|Aᵀ| = (2×4 − 3×5)
= 8 − 15
= −7

Verified

Q6.If |A| = 3, find |2A| (2×2 matrix)

Formula:

|kA| = k²|A|

|2A| = 2² × 3
= 4 × 3
= 12

Answer:12

Q7.If A is singular, can it have inverse? Explain.

Solution:

1.Singular means |A| = 0
2.Inverse exists only if determinant ≠ 0
3.So singular matrix has no inverse

Answer: No

FINAL REVISION SUMMARY

1.Order = rows × columns
2.Determinant formula = ad − bc
3.Inverse exists only if determinant ≠ 0
4.AB ≠ BA
5.|AB| = |A||B|
6.AA⁻¹ = I
7.Singular → determinant = 0
8.Non-Singular → determinant ≠ 0

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