MP Board Class 12 Maths Chapter 3: Matrices – Notes, Form…

📘 MP Board Class 12 Maths Chapter 3: Matrices – Notes, Formulas & Practice Questions

MP Board Class 12 Maths Chapter 3: Matrices (आव्यूह) is a fundamental chapter carrying 10–12 marks in the board exam. This chapter covers types of matrices, matrix operations (addition, subtraction, multiplication), transpose, symmetric and skew-symmetric matrices, elementary operations, and invertible matrices. Questions range from 1-mark MCQs to 4-mark and 6-mark long answer questions. Matrix methods are also used in solving systems of linear equations (Chapter 4: Determinants).

🔷 Introduction to Matrices – आव्यूह का परिचय

📌 Definition

A matrix is a rectangular array of numbers (real or complex) arranged in rows and columns. A matrix with m rows and n columns is called an m × n matrix (read as “m by n”).

General Form:
A = [aij]m×n =

⎡ a₁₁ a₁₂ … a₁ₙ ⎤
⎢ a₂₁ a₂₂ … a₂ₙ ⎥
⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎣ aₘ₁ aₘ₂ … aₘₙ ⎦

📝 Key Terminology

Term Meaning Example
Order Number of rows × columns 2×3 means 2 rows, 3 columns
Element (aij) Entry in i-th row, j-th column a₂₃ = element at row 2, col 3
Square Matrix Number of rows = columns (m = n) 3×3, 2×2 matrices
Row Matrix Only one row (1 × n) [1 2 3]
Column Matrix Only one column (m × 1) ⎡1⎤
⎢2⎥
⎣3⎦
🎯 Exam Tip: A matrix is denoted by capital letters A, B, C… and its elements by the corresponding lowercase letter with subscripts. The order (m×n) is always written as rows × columns — remember RC (Row then Column).

📊 Types of Matrices – आव्यूहों के प्रकार

Type Definition Example
Row Matrix 1 row only (1 × n) [5 -2 7 0]
Column Matrix 1 column only (m × 1) ⎡2⎤
⎢-1⎥
⎣3⎦
Square Matrix m = n (equal rows & columns) ⎡1 2⎤
⎣3 4⎦ (2×2)
Diagonal Matrix Square matrix with aij = 0 for i ≠ j ⎡2 0 0⎤
⎢0 5 0⎥
⎣0 0 -3⎦
Scalar Matrix Diagonal matrix with all diagonal entries equal ⎡3 0⎤
⎣0 3⎦
Identity Matrix (I) Scalar matrix with diagonal entries = 1 I₂ = ⎡1 0⎤
⎣0 1⎦
Zero Matrix (O) All elements = 0 ⎡0 0 0⎤
⎣0 0 0⎦
🎯 Exam Tip: The Identity Matrix I is the multiplicative identity — for any matrix A, A × I = I × A = A (provided the orders allow multiplication). This is frequently tested in 1-mark MCQs.

➕➖ Operations on Matrices – आव्यूह संक्रियाएँ

🔹 Addition of Matrices

Two matrices A and B can be added only if they have the same order. The sum is obtained by adding corresponding elements.

If A = [aij]m×n and B = [bij]m×n, then
A + B = [aij + bij]m×n

Properties of Matrix Addition:

  • Commutative: A + B = B + A
  • Associative: (A + B) + C = A + (B + C)
  • Identity: A + O = O + A = A (O = zero matrix)
  • Additive Inverse: A + (-A) = O

🔹 Scalar Multiplication

If k is a scalar (real number) and A = [aij], then:

kA = [k × aij] — each element is multiplied by k.

🔹 Subtraction of Matrices

A – B = [aij – bij]m×n — same order required.

✖️ Multiplication of Matrices – आव्यूहों का गुणन

📌 Condition for Multiplication

Two matrices A and B can be multiplied (AB) only if the number of columns in A = number of rows in B.

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

The element (AB)ij = Σk=1p aik × bkj

Properties of Matrix Multiplication:

  • NOT commutative in general: AB ≠ BA
  • Associative: (AB)C = A(BC)
  • Distributive: A(B + C) = AB + AC and (A + B)C = AC + BC
  • Multiplicative Identity: AI = IA = A (compatible identity matrix I)
  • AB = O does NOT imply A = O or B = O (null factor law fails)

🔄 Transpose of a Matrix – आव्यूह का परावर्त

📌 Definition

The transpose of a matrix A, denoted by A’ or AT, is obtained by interchanging its rows and columns.

If A = [aij]m×n, then AT = [aji]n×m

Example: If A = ⎡1 2 3⎤ then AT = ⎡1 4⎤
      ⎣4 5 6⎦    ⎢2 5⎥
          ⎣3 6⎦

📝 Properties of Transpose

  1. (AT)T = A — double transpose gives original
  2. (A + B)T = AT + BT
  3. (kA)T = kAT — for scalar k
  4. (AB)T = BTAT — NOTE the order reversal!

🪞 Symmetric & Skew-Symmetric Matrices

Property Symmetric Matrix Skew-Symmetric Matrix
Definition AT = A AT = -A
Condition aij = aji for all i, j aij = -aji for all i, j
Diagonal elements Any value Must be ZERO
Example ⎡1 2 3⎤
⎢2 4 5⎥
⎣3 5 6⎦		 ⎡0 2 -1⎤
⎢-2 0 3⎥
⎣1 -3 0⎦
🎯 Exam Tip: For any square matrix A, (A + AT) is symmetric and (A – AT) is skew-symmetric. Also, every square matrix can be uniquely expressed as: A = ½(A + AT) + ½(A – AT) — sum of symmetric and skew-symmetric parts. This is a very common 3-mark question!

🛠️ Elementary Operations (Transformations)

There are six elementary operations (three row operations and three column operations):

Notation Operation
Ri ↔ Rj Interchange i-th and j-th rows
Ri → kRi Multiply i-th row by a non-zero scalar k
Ri → Ri + kRj Add k times row j to row i

Similarly for column operations: Ci ↔ Cj, Ci → kCi, Ci → Ci + kCj.

These operations are used to find the inverse of a matrix using the Gauss-Jordan method (also called the elementary transformation method).

🔑 Invertible Matrices – व्युत्क्रमणीय आव्यूह

📌 Definition

A square matrix A of order n is said to be invertible if there exists a square matrix B of order n such that:

AB = BA = In

B is called the inverse of A, denoted by A-1.

🔑 Important Facts

  • Only square matrices can have inverses
  • A matrix is invertible iff |A| ≠ 0 (non-singular)
  • (A-1)-1 = A
  • (AB)-1 = B-1A-1 (note reversal!)
  • (AT)-1 = (A-1)T

🔑 How to Find Inverse using Elementary Row Operations

To find A-1 using row operations:

  1. Write A and I side by side: [A | I]
  2. Apply row operations to convert A on the left to I
  3. The matrix on the right becomes A-1

Symbolically: If we apply elementary row operations to A to get I, the same operations applied to I give A-1.

✏️ Solved Examples (5) – हल उदाहरण

Example 1: Matrix Addition

Question: If A = ⎡2 3⎤ and B = ⎡-1 5⎤, find A + B and 3A – 2B.
         ⎣4 0⎦    ⎣2 1⎦

Solution:

A + B = ⎡2+(-1) 3+5⎤ = ⎡1 8⎤
      ⎣4+2  0+1⎦ ⎣6 1⎦

3A – 2B = ⎡6 9⎤ + ⎡2 -10⎤ = ⎡6-2 9-(-10)⎤ = ⎡8 19⎤
      ⎣12 0⎦ ⎣-4 -2⎦ ⎣12-(-4) 0-(-2)⎦  ⎣16 2⎦

✅ Answer: A + B = ⎡1 8⎤, 3A – 2B = ⎡8 19⎤
        ⎣6 1⎦      ⎣16 2⎦

Example 2: Matrix Multiplication

Question: If A = ⎡1 2⎤ and B = ⎡5 6⎤, find AB and BA. Is AB = BA?
         ⎣3 4⎦    ⎣7 8⎦

Solution:

AB = ⎡1×5+2×7 1×6+2×8⎤ = ⎡5+14 6+16⎤ = ⎡19 22⎤
    ⎣3×5+4×7 3×6+4×8⎦ ⎣15+28 18+32⎦ ⎣43 50⎦

BA = ⎡5×1+6×3 5×2+6×4⎤ = ⎡5+18 10+24⎤ = ⎡23 34⎤
    ⎣7×1+8×3 7×2+8×4⎦ ⎣7+24 14+32⎦ ⎣31 46⎦

✅ AB ≠ BA, confirming that matrix multiplication is NOT commutative.

Example 3: Symmetric Matrix

Question: Show that the matrix A = ⎡1 2 -1⎤ can be expressed as the sum of a symmetric and a skew-symmetric matrix.
              ⎢-2 3 4⎥
              ⎣1 -2 0⎦

Solution:

First find AT = ⎡1 -2 1⎤
        ⎢2 3 -2⎥
        ⎣-1 4 0⎦

Symmetric part P = ½(A + AT):

A + AT = ⎡2 0 0⎤ → P = ½(A + AT) = ⎡1 0 0⎤
      ⎢0 6 2⎥        ⎢0 3 1⎥
      ⎣0 2 0⎦        ⎣0 1 0⎦

Skew-symmetric part Q = ½(A – AT):

A – AT = ⎡0 4 -2⎤ → Q = ½(A – AT) = ⎡0 2 -1⎤
      ⎢-4 0 6⎥        ⎢-2 0 3⎥
      ⎣2 -6 0⎦       ⎣1 -3 0⎦

✅ Verification: P + Q = A ✓

Example 4: Find Inverse using Elementary Transformation

Question: Find the inverse of A = ⎡2 3⎤ using elementary row operations.
             ⎣1 2⎦

Solution:

Write [A | I] = ⎡2 3 | 1 0⎤
         ⎣1 2 | 0 1⎦

R₁ ↔ R₂: ⎡1 2 | 0 1⎤
        ⎣2 3 | 1 0⎦

R₂ → R₂ – 2R₁: ⎡1 2 | 0 1⎤
         ⎣0 -1 | 1 -2⎦

R₂ → -R₂: ⎡1 2 | 0 1⎤
        ⎣0 1 | -1 2⎦

R₁ → R₁ – 2R₂: ⎡1 0 | 2 -3⎤
         ⎣0 1 | -1 2⎦

✅ A-1 = ⎡2 -3⎤
      ⎣-1 2⎦

Verification: A × A-1 = ⎡2 3⎤ × ⎡2 -3⎤ = ⎡1 0⎤ = I₂ ✓
           ⎣1 2⎦ ⎣-1 2⎦ ⎣0 1⎦

Example 5: Matrix Equation

Question: Solve the matrix equation for X: 2X + ⎡1 2⎤ = ⎡5 4⎤
              ⎣3 0⎦ ⎣1 2⎦

Solution:

2X = ⎡5 4⎤ – ⎡1 2⎤ = ⎡4 2⎤
   ⎣1 2⎦ ⎣3 0⎦ ⎣-2 2⎦

X = ½ × ⎡4 2⎤ = ⎡2 1⎤
    ⎣-2 2⎦ ⎣-1 1⎦

✅ Answer: X = ⎡2 1⎤
      ⎣-1 1⎦

📝 Practice Questions with Answers

Q1. Construct a 2×3 matrix whose elements are given by aij = (i + 2j)

👉 Click for Answer
Solution:
a₁₁ = 1+2 = 3, a₁₂ = 1+4 = 5, a₁₃ = 1+6 = 7
a₂₁ = 2+2 = 4, a₂₂ = 2+4 = 6, a₂₃ = 2+6 = 8

Required matrix: ⎡3 5 7⎤
        ⎣4 6 8⎦

Q2. If A = ⎡2 1⎤ and B = ⎡3 4⎤, find AB and BA. Comment on the result.
    ⎣0 -1⎦  ⎣-2 1⎦

👉 Click for Answer
AB = ⎡2×3+1×(-2) 2×4+1×1⎤ = ⎡4 9⎤
   ⎣0×3+(-1)×(-2) 0×4+(-1)×1⎦ ⎣2 -1⎦

BA = ⎡3×2+4×0 3×1+4×(-1)⎤ = ⎡6 -1⎤
   ⎣(-2)×2+1×0 (-2)×1+1×(-1)⎦ ⎣-4 -3⎦

AB ≠ BA — Matrix multiplication is not commutative.

Q3. If A = ⎡cos θ   sin θ⎤, show that ATA = I₂.
      ⎣-sin θ  cos θ⎦

👉 Click for Answer
AT = ⎡cos θ  -sin θ⎤
   ⎣sin θ   cos θ⎦

ATA = ⎡cos²θ + sin²θ   cosθ·sinθ – sinθ·cosθ⎤
    ⎣sinθ·cosθ – cosθ·sinθ   sin²θ + cos²θ⎦

ATA = ⎡1 0⎤ = I₂ ✓
    ⎣0 1⎦
This matrix represents a rotation in the plane — its transpose equals its inverse.

Q4. Find the inverse of A = ⎡3 10⎤ using elementary row operations.
           ⎣2 7⎦

👉 Click for Answer
[A | I] = ⎡3 10 | 1 0⎤
     ⎣2 7 | 0 1⎦

R₁ → R₁ – R₂: ⎡1 3 | 1 -1⎤
      ⎣2 7 | 0 1⎦

R₂ → R₂ – 2R₁: ⎡1 3 | 1 -1⎤
       ⎣0 1 | -2 3⎦

R₁ → R₁ – 3R₂: ⎡1 0 | 7 -10⎤
       ⎣0 1 | -2 3⎦

A-1 = ⎡7 -10⎤
     ⎣-2 3⎦

Verification: A × A-1 = ⎡3 10⎤ × ⎡7 -10⎤ = ⎡21-20   -30+30⎤ = ⎡1 0⎤ ✓
           ⎣2 7⎦ ⎣-2 3⎦ ⎣14-14   -20+21⎦ ⎣0 1⎦

Q5. If A = ⎡1 2⎤ and B = ⎡-1 4⎤, find X such that 2A + X = 3B.
      ⎣3 0⎦   ⎣2 1⎦

👉 Click for Answer
2A = ⎡2 4⎤
  ⎣6 0⎦

3B = ⎡-3 12⎤
  ⎣6 3⎦

X = 3B – 2A = ⎡-3-2   12-4⎤ = ⎡-5 8⎤
      ⎣6-6   3-0⎦  ⎣0 3⎦

Answer: X = ⎡-5 8⎤
     ⎣0 3⎦

Q6. For what value of x, the matrix A = ⎡0 2 -3⎤ is skew-symmetric?
               ⎢-2 0 4⎥
               ⎣3 -4 x⎦

👉 Click for Answer
For a skew-symmetric matrix, all diagonal elements must be ZERO.

Here the diagonal elements are: 0, 0, x.
For the matrix to be skew-symmetric: x = 0.

Also verify: a₁₂ = 2 and a₂₁ = -2 ✓
a₁₃ = -3 and a₃₁ = 3 ✓
a₂₃ = 4 and a₃₂ = -4 ✓

Answer: x = 0

Q7. If A = ⎡1 -1⎤ and B = ⎡2 3⎤, show that (AB)T = BTAT.
      ⎣2 0⎦   ⎣1 -2⎦

👉 Click for Answer
AB = ⎡1×2+(-1)×1   1×3+(-1)×(-2)⎤ = ⎡1 5⎤
  ⎣2×2+0×1   2×3+0×(-2)⎦  ⎣4 6⎦

(AB)T = ⎡1 4⎤
     ⎣5 6⎦

BTAT = ⎡2 1⎤ × ⎡1 2⎤ = ⎡2×1+1×2   2×(-1)+1×0⎤ = ⎡4 -2⎤ — Wait, let me recalculate.
     ⎣3 -2⎦ ⎣-1 0⎦ ⎣3×1+(-2)×2   3×(-1)+(-2)×0⎦ ⎣-1 -3⎦

Hmm, that doesn’t match. Let me redo this properly.

B = ⎡2 3⎤, so BT = ⎡2 1⎤
  ⎣1 -2⎦    ⎣3 -2⎦
A = ⎡1 -1⎤, so AT = ⎡1 2⎤
  ⎣2 0⎦    ⎣-1 0⎦

BTAT = ⎡2 1⎤ × ⎡1 2⎤
     ⎣3 -2⎦ ⎣-1 0⎦
= ⎡2×1+1×(-1)   2×2+1×0⎤ = ⎡1 4⎤
 ⎣3×1+(-2)×(-1)   3×2+(-2)×0⎦ ⎣5 6⎦

✅ (AB)T = BTAT = ⎡1 4⎤ — Verified!
         ⎣5 6⎦

Q8. If A = ⎡3 1⎤, find k such that A² = 5A + kI.
      ⎣-1 2⎦

👉 Click for Answer
A² = A × A = ⎡3 1⎤ × ⎡3 1⎤ = ⎡3×3+1×(-1)   3×1+1×2⎤ = ⎡8 5⎤
     ⎣-1 2⎦ ⎣-1 2⎦ ⎣(-1)×3+2×(-1)   (-1)×1+2×2⎦ ⎣-5 3⎦

5A = ⎡15 5⎤
  ⎣-5 10⎦

A² = 5A + kI
⎡8 5⎤ = ⎡15 5⎤ + k⎡1 0⎤
⎣-5 3⎦ ⎣-5 10⎦  ⎣0 1⎦

Comparing (1,1): 8 = 15 + k → k = -7
Comparing (2,2): 3 = 10 + k → k = -7 ✓

Answer: k = -7

📌 Important Formulas – Quick Revision

# Formula / Property
1 Addition: (A + B)ij = aij + bij (same order required)
2 Scalar multiplication: (kA)ij = k × aij
3 Multiplication: (AB)ij = Σk aik · bkj
4 Transpose: (AT)ij = aji
5 Transpose properties: (A+B)T = AT+BT, (AB)T = BTAT, (kA)T = kAT
6 Symmetric: AT = A (aij = aji)
7 Skew-symmetric: AT = -A (aij = -aji, diagonal = 0)
8 Inverse: A-1 exists iff |A| ≠ 0
9 Inverse properties: (AB)-1 = B-1A-1, (A-1)-1 = A, (AT)-1 = (A-1)T
10 Symmetric + Skew-symmetric: A = ½(A+AT) + ½(A-AT)
📖 Key Takeaway: Matrices form the foundation for solving systems of linear equations (Chapter 4). Master these concepts — especially matrix multiplication, transpose properties, and finding inverses — as they’ll be used extensively in Determinants and later in Linear Programming.

📅 Generated: 30 July 2026 | 📚 MP Board Class 12 Maths — Chapter 3: Matrices (आव्यूह)

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