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Matrices
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Class 12MathsChapter 3NCERTCBSE
Matrices – Class 12 Maths Chapter 3
📌 Quick Overview: Chapter 3 introduces matrices — rectangular arrangements of numbers. You will learn types of matrices, matrix operations (addition, subtraction, multiplication), transpose, symmetric and skew-symmetric matrices, and elementary row operations. This chapter is directly connected to Chapter 4 (Determinants) and is heavily tested in board exams.
Topics Covered:
- > What is a Matrix?
- > Order of a Matrix
- > Types of Matrices
- > Matrix Addition
- > Scalar Multiplication
- > Matrix Multiplication
- > Transpose of a Matrix
- > Symmetric Matrix
- > Skew-Symmetric Matrix
- > Invertible Matrices
- > Elementary Operations
- > Important Board Questions
1. What is a Matrix?
Definition: A matrix is a rectangular arrangement of numbers (real or complex) enclosed in brackets, arranged in rows and columns.
A matrix with m rows and n columns is called an m x n matrix (read as "m by n matrix"). The numbers in the matrix are called elements or entries.
A matrix with m rows and n columns is called an m x n matrix (read as "m by n matrix"). The numbers in the matrix are called elements or entries.
Example: A = [2, 3, 5 / 1, 4, 7] is a 2 x 3 matrix (2 rows, 3 columns).
General notation: A = [aij]m x n, where aij = element in i-th row and j-th column.
General notation: A = [aij]m x n, where aij = element in i-th row and j-th column.
⭐ Order of a Matrix: If a matrix has m rows and n columns, its order is m x n and it has m x n total elements.
Example: A 3 x 4 matrix has 3 x 4 = 12 elements.
Example: A 3 x 4 matrix has 3 x 4 = 12 elements.
2. Types of Matrices
| Type | Definition | Example (Order) |
|---|---|---|
| Row Matrix | Only 1 row | [1, 2, 3] (1 x 3) |
| Column Matrix | Only 1 column | [1 / 2 / 3] (3 x 1) |
| Square Matrix | Number of rows = number of columns (m = n) | 2 x 2, 3 x 3 |
| Zero Matrix (Null) | All elements are 0 | [0, 0 / 0, 0] |
| Diagonal Matrix | Square matrix; non-diagonal elements = 0 | [2, 0 / 0, 5] |
| Scalar Matrix | Diagonal matrix with all diagonal elements equal | [3, 0 / 0, 3] |
| Identity Matrix (I) | Diagonal matrix with all diagonal elements = 1 | [1, 0 / 0, 1] |
| Upper Triangular | All elements below diagonal = 0 | [1, 2 / 0, 3] |
| Lower Triangular | All elements above diagonal = 0 | [1, 0 / 3, 4] |
⭐ Equality of Matrices: Two matrices A and B are equal (A = B) if and only if:
(i) They have the same order, AND
(ii) Their corresponding elements are equal: aij = bij for all i, j.
(i) They have the same order, AND
(ii) Their corresponding elements are equal: aij = bij for all i, j.
3. Operations on Matrices
3.1 Addition of Matrices
Rule: Two matrices can be added only if they have the same order. Add corresponding elements.
If A = [aij] and B = [bij] are both m x n matrices, then A + B = [aij + bij].
If A = [aij] and B = [bij] are both m x n matrices, then A + B = [aij + bij].
Example:
A = [1, 2 / 3, 4], B = [5, 6 / 7, 8]
A + B = [1+5, 2+6 / 3+7, 4+8] = [6, 8 / 10, 12]
A = [1, 2 / 3, 4], B = [5, 6 / 7, 8]
A + B = [1+5, 2+6 / 3+7, 4+8] = [6, 8 / 10, 12]
Properties of Matrix Addition:
1. Commutative: A + B = B + A
2. Associative: (A + B) + C = A + (B + C)
3. Identity: A + O = O + A = A (O = zero matrix)
4. Additive Inverse: A + (-A) = O
3.2 Scalar Multiplication
Rule: Multiply every element of the matrix by the scalar (constant) k.
kA = [k * aij]
kA = [k * aij]
Example: If A = [1, 2 / 3, 4] and k = 3,
3A = [3, 6 / 9, 12]
3A = [3, 6 / 9, 12]
3.3 Multiplication of Matrices ⭐
Rule: Matrix A (m x n) can be multiplied with matrix B (n x p) ONLY when the number of columns of A = number of rows of B. The result AB has order m x p.
Element in i-th row, j-th column of AB = sum of products of i-th row of A with j-th column of B.
(AB)ij = ai1b1j + ai2b2j + ... + ainbnj
Element in i-th row, j-th column of AB = sum of products of i-th row of A with j-th column of B.
(AB)ij = ai1b1j + ai2b2j + ... + ainbnj
Example: A = [1, 2 / 3, 4] (2x2), B = [5, 6 / 7, 8] (2x2)
AB = [(1x5 + 2x7), (1x6 + 2x8) / (3x5 + 4x7), (3x6 + 4x8)]
= [(5+14), (6+16) / (15+28), (18+32)]
= [19, 22 / 43, 50]
AB = [(1x5 + 2x7), (1x6 + 2x8) / (3x5 + 4x7), (3x6 + 4x8)]
= [(5+14), (6+16) / (15+28), (18+32)]
= [19, 22 / 43, 50]
Properties of Matrix Multiplication:
1. NOT commutative: AB is NOT equal to BA (in general)
2. Associative: (AB)C = A(BC)
3. Distributive: A(B + C) = AB + AC
4. Identity: AI = IA = A
5. Zero matrix: A.O = O.A = O
6. If AB = O, it does NOT mean A = O or B = O (unlike numbers!)
⚠️ Critical Warning: In matrices, AB = O does NOT imply A = O or B = O. Also, AB = AC does NOT imply B = C (cancellation law does NOT hold).
4. Transpose of a Matrix
Definition: The transpose of a matrix A is obtained by interchanging its rows and columns. It is denoted by A' or AT.
If A = [aij]m x n, then A' = [aji]n x m.
If A = [aij]m x n, then A' = [aji]n x m.
Example: A = [1, 2, 3 / 4, 5, 6] (2 x 3)
A' = [1, 4 / 2, 5 / 3, 6] (3 x 2)
A' = [1, 4 / 2, 5 / 3, 6] (3 x 2)
Properties of Transpose:
1. (A')' = A
2. (A + B)' = A' + B'
3. (kA)' = kA'
4. (AB)' = B'A' (ORDER REVERSES!)
⭐ Key Property: (AB)' = B'A' — the order reverses when transposing a product. This is tested frequently.
5. Symmetric and Skew-Symmetric Matrices ⭐
5.1 Symmetric Matrix
Definition: A square matrix A is called symmetric if A' = A.
i.e., aij = aji for all i, j.
i.e., aij = aji for all i, j.
Example: A = [1, 2, 3 / 2, 5, 6 / 3, 6, 9]
Here A' = A, so A is symmetric.
Here A' = A, so A is symmetric.
5.2 Skew-Symmetric Matrix
Definition: A square matrix A is called skew-symmetric if A' = -A.
i.e., aij = -aji for all i, j.
This means all diagonal elements must be 0 (since aii = -aii implies aii = 0).
i.e., aij = -aji for all i, j.
This means all diagonal elements must be 0 (since aii = -aii implies aii = 0).
Example: A = [0, 2, -3 / -2, 0, 5 / 3, -5, 0]
Here A' = -A, so A is skew-symmetric. Notice all diagonal elements = 0.
Here A' = -A, so A is skew-symmetric. Notice all diagonal elements = 0.
⭐ Most Important Theorem: Any square matrix A can be expressed as the sum of a symmetric and a skew-symmetric matrix:
A = (A + A')/2 + (A - A')/2
where (A + A')/2 is symmetric and (A - A')/2 is skew-symmetric.
A = (A + A')/2 + (A - A')/2
where (A + A')/2 is symmetric and (A - A')/2 is skew-symmetric.
Example: Express A = [3, 5 / 1, -1] as sum of symmetric and skew-symmetric matrix.
A' = [3, 1 / 5, -1]
Symmetric part P = (A + A')/2 = ([3,5/1,-1] + [3,1/5,-1])/2 = [6,6/6,-2]/2 = [3,3/3,-1]
Skew-symmetric part Q = (A - A')/2 = ([3,5/1,-1] - [3,1/5,-1])/2 = [0,4/-4,0]/2 = [0,2/-2,0]
Check: P + Q = [3,3/3,-1] + [0,2/-2,0] = [3,5/1,-1] = A ✓
A' = [3, 1 / 5, -1]
Symmetric part P = (A + A')/2 = ([3,5/1,-1] + [3,1/5,-1])/2 = [6,6/6,-2]/2 = [3,3/3,-1]
Skew-symmetric part Q = (A - A')/2 = ([3,5/1,-1] - [3,1/5,-1])/2 = [0,4/-4,0]/2 = [0,2/-2,0]
Check: P + Q = [3,3/3,-1] + [0,2/-2,0] = [3,5/1,-1] = A ✓
6. Elementary Row and Column Operations
These are operations used to simplify matrices (especially to find inverse). There are 3 types:
| Operation | Notation | Meaning |
|---|---|---|
| Interchange two rows/cols | Ri ↔ Rj | Swap row i and row j |
| Multiply a row/col by k ≠ 0 | Ri → kRi | Multiply every element of row i by k |
| Add multiple of one row to another | Ri → Ri + kRj | Replace row i with row i + k times row j |
7. Invertible Matrix
Definition: A square matrix A of order n is called invertible if there exists a matrix B such that:
AB = BA = In
Then B is called the inverse of A, written as A-1.
AB = BA = In
Then B is called the inverse of A, written as A-1.
⭐ Key Facts:
> A matrix has an inverse ONLY if it is a square matrix.
> The inverse is unique (no matrix can have two different inverses).
> (A-1)-1 = A
> (AB)-1 = B-1A-1 (order reverses!)
> (A')-1 = (A-1)'
> A matrix has an inverse ONLY if it is a square matrix.
> The inverse is unique (no matrix can have two different inverses).
> (A-1)-1 = A
> (AB)-1 = B-1A-1 (order reverses!)
> (A')-1 = (A-1)'
Finding Inverse using Elementary Row Operations
Write [A | I], then apply row operations to convert A to I. The right side becomes A-1.
i.e., [A | I] → [I | A-1]
i.e., [A | I] → [I | A-1]
Example: Find the inverse of A = [1, 2 / 3, 7] using elementary row operations.
[A|I] = [1, 2 | 1, 0 / 3, 7 | 0, 1]
R2 → R2 - 3R1:
[1, 2 | 1, 0 / 0, 1 | -3, 1]
R1 → R1 - 2R2:
[1, 0 | 7, -2 / 0, 1 | -3, 1]
∴ A-1 = [7, -2 / -3, 1]
Verify: AA-1 = [1,2/3,7][7,-2/-3,1] = [7-6,-2+2/21-21,-6+7] = [1,0/0,1] = I ✓
[A|I] = [1, 2 | 1, 0 / 3, 7 | 0, 1]
R2 → R2 - 3R1:
[1, 2 | 1, 0 / 0, 1 | -3, 1]
R1 → R1 - 2R2:
[1, 0 | 7, -2 / 0, 1 | -3, 1]
∴ A-1 = [7, -2 / -3, 1]
Verify: AA-1 = [1,2/3,7][7,-2/-3,1] = [7-6,-2+2/21-21,-6+7] = [1,0/0,1] = I ✓
8. Important Board Exam Questions
Q1. If A = [0, -tan(x/2) / tan(x/2), 0] and I is the identity matrix of order 2, show that I + A = (I - A)[cos x, -sin x / sin x, cos x].
This is a standard proof-type question. Compute LHS and RHS separately using the given matrix A, using the identities cos x = (1-tan²(x/2))/(1+tan²(x/2)) and sin x = 2tan(x/2)/(1+tan²(x/2)).
Q2. Find x, y, z if [x+y+z / x+z / y+z] = [9 / 5 / 7].
From equality of matrices:
x + y + z = 9 ... (i)
x + z = 5 ... (ii)
y + z = 7 ... (iii)
From (i) - (ii): y = 4
From (iii): z = 7 - 4 = 3
From (ii): x = 5 - 3 = 2
∴ x = 2, y = 4, z = 3
x + y + z = 9 ... (i)
x + z = 5 ... (ii)
y + z = 7 ... (iii)
From (i) - (ii): y = 4
From (iii): z = 7 - 4 = 3
From (ii): x = 5 - 3 = 2
∴ x = 2, y = 4, z = 3
Q3. If A = [3, -4 / 1, -1], prove that An = [1+2n, -4n / n, 1-2n] using mathematical induction.
Step 1 (Base case n=1): A1 = [1+2, -4 / 1, 1-2] = [3, -4 / 1, -1] = A ✓
Step 2 (Assume true for n=k): Ak = [1+2k, -4k / k, 1-2k]
Step 3 (Prove for n=k+1):
Ak+1 = Ak.A = [1+2k, -4k / k, 1-2k].[3,-4/1,-1]
= [3+6k-4k, -4-8k+4k / 3k+1-2k, -4k-1+2k]
= [3+2k, -4-4k / k+1, -1-2k]
= [1+2(k+1), -4(k+1) / (k+1), 1-2(k+1)] ✓
Hence proved by induction.
Step 2 (Assume true for n=k): Ak = [1+2k, -4k / k, 1-2k]
Step 3 (Prove for n=k+1):
Ak+1 = Ak.A = [1+2k, -4k / k, 1-2k].[3,-4/1,-1]
= [3+6k-4k, -4-8k+4k / 3k+1-2k, -4k-1+2k]
= [3+2k, -4-4k / k+1, -1-2k]
= [1+2(k+1), -4(k+1) / (k+1), 1-2(k+1)] ✓
Hence proved by induction.
Q4. Express A = [6, 2 / -2, 3] as a sum of a symmetric and a skew-symmetric matrix.
A' = [6, -2 / 2, 3]
Symmetric P = (A + A')/2 = [12, 0 / 0, 6]/2 = [6, 0 / 0, 3]
Skew-symmetric Q = (A - A')/2 = [0, 4 / -4, 0]/2 = [0, 2 / -2, 0]
Check: P + Q = [6,0/0,3] + [0,2/-2,0] = [6,2/-2,3] = A ✓
Symmetric P = (A + A')/2 = [12, 0 / 0, 6]/2 = [6, 0 / 0, 3]
Skew-symmetric Q = (A - A')/2 = [0, 4 / -4, 0]/2 = [0, 2 / -2, 0]
Check: P + Q = [6,0/0,3] + [0,2/-2,0] = [6,2/-2,3] = A ✓
Q5. If A and B are symmetric matrices of the same order, prove that AB - BA is skew-symmetric.
Given: A' = A and B' = B.
Let C = AB - BA.
C' = (AB - BA)' = (AB)' - (BA)'
= B'A' - A'B' = BA - AB = -(AB - BA) = -C
∴ C' = -C, so AB - BA is skew-symmetric. ✓
Let C = AB - BA.
C' = (AB - BA)' = (AB)' - (BA)'
= B'A' - A'B' = BA - AB = -(AB - BA) = -C
∴ C' = -C, so AB - BA is skew-symmetric. ✓
9. Key Formulas at a Glance
MATRIX BASICS:
Order m x n --> m x n total elements
(AB)_ij = sum of (row i of A) x (col j of B)
AB exists only if cols of A = rows of B
TRANSPOSE:
(A')' = A
(A+B)' = A' + B'
(AB)' = B'A' [order reverses]
SYMMETRIC: A' = A
SKEW-SYM: A' = -A (diagonal elements = 0)
ANY MATRIX: A = (A+A')/2 + (A-A')/2
[symmetric] [skew-symmetric]
INVERSE:
(AB)^-1 = B^-1 . A^-1 [order reverses]
(A')^-1 = (A^-1)'
(A^-1)^-1 = A
AA^-1 = A^-1A = I
10. MCQ Practice (1 Mark)
1. If A is a matrix of order 3 x 4 and B is a matrix of order 4 x 3, then order of AB is:
(a) 4 x 4 (b) 3 x 3 (c) 4 x 3 (d) 3 x 4
Answer: (b) 3 x 3
2. If A is a symmetric matrix, then A³ is:
(a) Skew-symmetric (b) Symmetric (c) Zero matrix (d) Identity matrix
Answer: (b) Symmetric — because (A³)' = (A')³ = A³
3. If A is a skew-symmetric matrix of odd order, then |A| is:
(a) 1 (b) Non-zero (c) 0 (d) None
Answer: (c) 0
4. For matrices A and B, (AB)' equals:
(a) A'B' (b) B'A' (c) AB (d) BA
Answer: (b) B'A'
11. Exam Tips
- For multiplication, always check order compatibility first — cols of A must equal rows of B.
- AB ≠ BA in general — never cancel matrices like numbers.
- Diagonal elements of a skew-symmetric matrix are ALWAYS zero — this fact alone appears in MCQs.
- To prove AB - BA is skew-symmetric: find (AB-BA)' and show it equals -(AB-BA).
- For inverse by row operations: write [A|I] and reduce left side to I — right side becomes A⁻¹.
- Always verify your inverse: multiply AA⁻¹ and check you get the identity matrix.
- The order reversal rule (AB)' = B'A' and (AB)⁻¹ = B⁻¹A⁻¹ is tested every year.
✅ Summary: Matrices is a foundational chapter — master the multiplication rule, transpose properties, and the symmetric/skew-symmetric theorem. These topics directly feed into Chapter 4 (Determinants) which comes next. Practice the "express as sum" and "prove" type questions as they appear consistently in CBSE board exams.
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