CLASS XII MATHEMATICS CHAPTER 3

Matrices

Complete Class XII Mathematics chapter notes covering matrix notation, order, types of matrices, matrix operations, transpose, symmetric and skew-symmetric matrices, elementary transformations, inverse of a matrix and matrix equations.

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MATRIX REPRESENTATION

Introduction to Matrices

A matrix is a rectangular arrangement of numbers, symbols or expressions in rows and columns.

General Form of a Matrix

A matrix having m rows and n columns is called an m × n matrix.

a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃

A = [a_ij]_{m × n}

Example

Consider the matrix:

2 −1 3 0 4 5

Order = 2 × 3

It contains 2 rows and 3 columns.

Visualising Matrix Elements

Matrix Basics

Fundamental concepts required for matrix problems.

Order

Order = Number of rows × Number of columns.

3 × 4

Element

a_ij denotes the element in the i^th row and j^th column.

a₂₃

Equality of Matrices

Two matrices are equal if they have the same order and corresponding elements are equal.

Important Rule

Before performing an operation, always check the order of the matrices.

Types of Matrices

Important matrix types for Class XII.

Row Matrix

Exactly one row.

2 5 −1

Column Matrix

Exactly one column.

3 −2 4

Rectangular Matrix

Rows and columns are unequal.

1 2 3 4 5 6

Square Matrix

Rows = columns.

1 2 3 4

Zero Matrix

Every element is zero.

0 0 0 0

Diagonal Matrix

Off-diagonal entries are zero.

4 0 0 0 −2 0 0 0 5

Scalar Matrix

Equal diagonal elements.

3 0 0 3

Identity Matrix

Diagonal elements are 1.

1 0 0 0 1 0 0 0 1

Upper Triangular

Entries below diagonal are zero.

2 4 5 0 3 6 0 0 7

Lower Triangular

Entries above diagonal are zero.

2 0 0 4 3 0 5 6 7

Matrix Operations

Addition, subtraction, scalar multiplication and matrix multiplication.

Addition of Matrices

Matrices of the same order can be added by adding their corresponding elements.

2 3 −1 4

5 −2 3 1

A + B

7 1 2 5

Scalar Multiplication

Every element of a matrix is multiplied by the scalar.

3 ×

1 2 3 4

3 6 9 12

Matrix Multiplication

Multiplication Condition

If A is an m × n matrix and B is an n × p matrix, then AB exists and is an m × p matrix.

1 2 3 4

2 0 1 5

4 10 10 20

(AB)_ij = Σ a_ikb_kj

Matrix Multiplication is Not Generally Commutative

4 10 10 20

≠

2 4 16 22

Therefore, AB ≠ BA in general.

Transpose of a Matrix

The transpose changes rows into columns.

1 2 3 4 5 6

⇩

A^T

1 4 2 5 3 6

A = [a_ij] ⇒ A^T = [a_ji]

Important Properties

(A^T)^T = A

(A + B)^T = A^T + B^T

(kA)^T = kA^T

(AB)^T = B^TA^T

Exam Alert

While taking the transpose of a product, reverse the order of the matrices.

(AB)^T = B^TA^T

Symmetric & Skew-Symmetric Matrices

Special square matrices identified through transpose.

Symmetric Matrix

A^T = A

2 3 1 3 5 −2 1 −2 4

Corresponding elements on opposite sides of the principal diagonal are equal.

Skew-Symmetric Matrix

A^T = −A

0 4 −1 −4 0 2 1 −2 0

Every diagonal element of a skew-symmetric matrix is zero.

Decomposition of a Square Matrix

Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

A = ½(A + A^T) + ½(A − A^T)

Symmetric Part

A_s = ½(A + A^T)

Skew-Symmetric Part

A_k = ½(A − A^T)

Elementary Transformations

Elementary operations are useful for matrix simplification and finding inverses.

Interchange

R_i ↔ R_j

Interchange two rows.

Scaling

R_i → kR_i

k ≠ 0.

Row Replacement

R_i → R_i + kR_j

Add a multiple of another row.

Elementary Transformation Flow

[ A | I ] → [ I | A⁻¹ ]

Inverse of a Matrix

The inverse matrix is used to solve matrix equations.

AA⁻¹ = A⁻¹A = I

Inverse of a 2 × 2 Matrix

a b c d

then, provided ad − bc ≠ 0,

d −b −c a

× 1 / (ad − bc)

Visual: Matrix × Inverse = Identity

Solving Linear Equations Using Matrices

A system of linear equations can be written in matrix form.

AX = B
↓
A⁻¹AX = A⁻¹B
X = A⁻¹B

Example

Solve:

2x + y = 5
x − y = 1

Write the equations as AX = B.

2 1 1 −1

x y

5 1

|A| = (2)(−1) − (1)(1) = −3 ≠ 0

X = A⁻¹B

x = 2 and y = 1

Solved Examples

Selected examination-oriented examples.

Example 1: Find the order and a₂₃

Given:

2 5 7 1 4 8 3 6 9

There are 3 rows and 3 columns.

Therefore, order = 3 × 3.

The element in the second row and third column is 8.
Therefore, a₂₃ = 8.

Example 2: Find x and y using equality of matrices

Given:

x 3 4 y

5 3 4 7

Corresponding elements are equal.

x = 5, y = 7

Example 3: Add two matrices

2 3 4 5

1 6 2 3

3 9 6 8

Example 4: Find 3A

1 2 3 4

then

3A =

3 6 9 12

Example 5: Find AB

1 2 3 4

2 0 1 5

4 10 10 20

Example 6: Find the transpose

1 2 3 4 5 6

→

1 4 2 5 3 6

Example 7: Check whether a matrix is symmetric

2 3 3 5

Its transpose is the same matrix.

A^T = A

Hence, A is symmetric.

Example 8: Check whether a matrix is skew-symmetric

0 4 −4 0

A^T = −A

Hence, A is skew-symmetric.

Example 9: Find the inverse of a 2 × 2 matrix

Let

2 3 1 2

Determinant:

|A| = (2)(2) − (3)(1) = 1

Therefore,

2 −3 −1 2

is A⁻¹.

Example 10: Solve equations using inverse matrix

Given:

2x + y = 5
x − y = 1

Write:

AX = B

Since

|A| = −3 ≠ 0

A is invertible.

Hence:

X = A⁻¹B

Therefore:

x = 2, y = 1

Example 11: Inverse of a 3 × 3 matrix

Consider:

1 2 3 0 1 4 5 6 0

Using elementary transformations on

[ A | I ]

we obtain

−24 18 5 20 −15 −4 −5 4 1

Hence this is A⁻¹.

Example 12: Matrix decomposition

For any square matrix A:

A = ½(A + A^T) + ½(A − A^T)

The first part is symmetric and the second part is skew-symmetric.

Practice Questions

Practice important concepts from Chapter 3.

Q1. Define a matrix and explain its order.

Answer: A matrix is a rectangular arrangement of elements in rows and columns. Its order is the number of rows × number of columns.

Q2. Find the order and a₂₃ of [ [2,4,6], [1,3,5], [7,8,9] ].

Answer: Order = 3 × 3 and a₂₃ = 5.

Q3. When can two matrices be added?

Answer: Only matrices having the same order can be added.

Q4. Find 2A if A = [ [1,2], [3,4] ].

Answer: 2A = [ [2,4], [6,8] ].

Q5. Find AB for A = [ [1,2], [3,4] ] and B = [ [2,0], [1,5] ].

Answer: AB = [ [4,10], [10,20] ].

Q6. Is matrix multiplication generally commutative?

Answer: No. In general AB ≠ BA.

Q7. Find the transpose of [ [1,2,3], [4,5,6] ].

Answer: Aᵀ = [ [1,4], [2,5], [3,6] ].

Q8. State the condition for a matrix to be symmetric.

Answer: Aᵀ = A.

Q9. State the condition for a matrix to be skew-symmetric.

Answer: Aᵀ = −A.

Q10. What are the diagonal elements of a skew-symmetric matrix?

Answer: All diagonal elements are zero.

Q11. State the three elementary row operations.

Answer: Interchange rows, multiply a row by a non-zero scalar, and add a multiple of one row to another.

Q12. Write the formula for the inverse of a 2 × 2 matrix.

Answer: For A = [ [a,b], [c,d] ], A⁻¹ = 1/(ad−bc) [ [d,−b], [−c,a] ], provided ad−bc ≠ 0.

Q13. When does a square matrix have an inverse?

Answer: When its determinant is non-zero.

Q14. If AX = B and A is invertible, find X.

Answer: X = A⁻¹B.

Q15. Write the transpose property of a product.

Answer: (AB)ᵀ = BᵀAᵀ.

Q16. Write the inverse property of a product.

Answer: (AB)⁻¹ = B⁻¹A⁻¹.

Q17. Can AB exist when BA does not exist?

Answer: Yes. Matrix multiplication depends on compatible dimensions.

Q18. What is an identity matrix?

Answer: A square matrix having 1 on the main diagonal and 0 elsewhere.

Q19. Write the symmetric and skew-symmetric decomposition of A.

Answer: A = ½(A + Aᵀ) + ½(A − Aᵀ).

Q20. Why is matrix multiplication not generally commutative?

Answer: Because changing the order of the matrices generally changes the row-column products and therefore the result.

HOT Questions

Higher-order thinking questions for examination preparation.

HOT 1

Show that every square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

HOT 2

Prove that (AB)ᵀ = BᵀAᵀ.

HOT 3

If A and B are invertible matrices, prove that (AB)⁻¹ = B⁻¹A⁻¹.

HOT 4

Solve a system of three linear equations using the inverse matrix method.

HOT 5

Find the values of parameters for which a given matrix becomes symmetric.

HOT 6

If A and B are symmetric matrices, determine the condition under which AB is symmetric.

HOT 7

Prove that every diagonal element of a skew-symmetric matrix is zero.

HOT 8

If AX = B and A is invertible, derive X = A⁻¹B.

HOT 9

Find X from AX + C = B when A is invertible.

HOT 10

Explain why the cancellation law is not generally valid for matrices.

Quick Revision

Important formulas and concepts at a glance.

Concept	Formula / Rule
Order	m × n
Addition	Same order required
Multiplication	Columns of first = rows of second
Transpose	(Aᵀ)ᵀ = A
Product Transpose	(AB)ᵀ = BᵀAᵀ
Symmetric	Aᵀ = A
Skew-Symmetric	Aᵀ = −A
Skew diagonal elements	a_ii = 0
Decomposition	A = ½(A+Aᵀ)+½(A−Aᵀ)
Inverse	AA⁻¹ = A⁻¹A = I
Product Inverse	(AB)⁻¹ = B⁻¹A⁻¹
Matrix Equation	AX = B ⇒ X = A⁻¹B

Matrices Chapter Concept Map

Exam Tips

Important points to remember while solving questions.

✓ Check Matrix Order

Check dimensions before multiplication.

✓ Remember AB ≠ BA

Matrix multiplication is not generally commutative.

✓ Reverse Order in Transpose

(AB)ᵀ = BᵀAᵀ

✓ Check Inverse Condition

|A| ≠ 0

✓ Skew-Symmetric Diagonal

All diagonal elements are zero.

✓ Verify the Inverse

AA⁻¹ = I

Chapter 3 Checklist

□ Matrix notation

□ Order of matrix

□ Types of matrices

□ Addition

□ Matrix multiplication

□ Transpose

□ Symmetric matrices

□ Skew-symmetric matrices

□ Elementary transformations

□ Inverse

□ Matrix equations

□ Practice questions

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