Class 12 Mathematics Notes Chapter 3 (Matrices) – Mathematics Part-I Book

Alright class, let's dive into Chapter 3: Matrices. This is a fundamental topic, not just for your board exams but also frequently tested in various government recruitment exams. Pay close attention to the definitions and properties.

Chapter 3: Matrices - Detailed Notes for Government Exam Preparation

1. Definition and Notation

• Matrix: A matrix is an ordered rectangular array (arrangement) of numbers or functions. These numbers or functions are called the elements or entries of the matrix.
• Notation: Matrices are usually denoted by capital letters (e.g., A, B, C). The elements are enclosed in square brackets [] or parentheses ().
  • Example: A = [ a_ij ]_(m x n)
• Order of a Matrix: A matrix having m rows and n columns is called a matrix of order m × n (read as 'm by n').
  • The total number of elements in an m × n matrix is mn.
  • a_ij represents the element in the i-th row and j-th column.

2. Types of Matrices

• (i) Column Matrix: A matrix having only one column (n=1). Order: m × 1.
  • Example: [ 1 ]
[ 2 ]
[ 3 ] is a 3 × 1 column matrix.
• (ii) Row Matrix: A matrix having only one row (m=1). Order: 1 × n.
  • Example: [ 1 5 9 ] is a 1 × 3 row matrix.
• (iii) Square Matrix: A matrix in which the number of rows equals the number of columns (m=n). Order: n × n or simply n.
  • Example: [ 1 2 ]
[ 3 4 ] is a square matrix of order 2.
  • Principal Diagonal: In a square matrix A = [a_ij]_(n x n), the elements a_11, a_22, ..., a_nn constitute the principal diagonal (or main diagonal).
• (iv) Diagonal Matrix: A square matrix where all non-diagonal elements are zero (a_ij = 0 for i ≠ j).
  • Example: [ 2 0 0 ]
[ 0 -1 0 ]
[ 0 0 5 ]
• (v) Scalar Matrix: A diagonal matrix where all diagonal elements are equal (a_ij = 0 for i ≠ j, and a_ii = k for some constant k).
  • Example: [ 7 0 0 ]
[ 0 7 0 ]
[ 0 0 7 ]
• (vi) Identity Matrix (Unit Matrix): A square matrix where all diagonal elements are 1 and all non-diagonal elements are 0 (a_ij = 1 if i = j, a_ij = 0 if i ≠ j). Denoted by I or I_n (identity matrix of order n).
  • Example: I_2 = [ 1 0 ], I_3 = [ 1 0 0 ]
[ 0 1 ] [ 0 1 0 ]
[ 0 0 1 ]
  • I acts as the multiplicative identity for matrix multiplication (AI = IA = A).
• (vii) Zero Matrix (Null Matrix): A matrix where all elements are zero. Denoted by O.
  • Example: O = [ 0 0 ]
[ 0 0 ]
  • O acts as the additive identity for matrix addition (A + O = O + A = A).

3. Equality of Matrices

Two matrices A = [a_ij] and B = [b_ij] are equal if:

• (i) They are of the same order (m × n).
• (ii) Each corresponding element is equal (a_ij = b_ij for all i and j).

4. Operations on Matrices

• (i) Addition of Matrices:
  • Condition: Matrices must be of the same order.
  • Operation: If A = [a_ij] and B = [b_ij] are matrices of order m × n, then their sum A + B is a matrix C = [c_ij] of order m × n, where c_ij = a_ij + b_ij for all i and j.
  • Properties:
    • Commutative Law: A + B = B + A
    • Associative Law: (A + B) + C = A + (B + C)
    • Existence of Additive Identity: A + O = O + A = A (where O is the zero matrix of the same order as A)
    • Existence of Additive Inverse: For every matrix A, there exists a matrix -A such that A + (-A) = (-A) + A = O. -A is obtained by multiplying each element of A by -1.
• (ii) Multiplication of a Matrix by a Scalar:
  • Operation: If A = [a_ij] is an m × n matrix and k is a scalar (a number), then kA is another m × n matrix obtained by multiplying each element of A by k. kA = [k * a_ij].
  • Properties:
    • k(A + B) = kA + kB
    • (k + l)A = kA + lA (where k and l are scalars)
• (iii) Multiplication of Matrices:
  • Condition: For the product AB to be defined, the number of columns in matrix A must be equal to the number of rows in matrix B.
  • If A is of order m × n and B is of order n × p, then the product AB is a matrix C of order m × p.
  • Operation: The element c_ij (element in the i-th row and j-th column of AB) is obtained by multiplying the elements of the i-th row of A with the corresponding elements of the j-th column of B and then summing the products.
    c_ij = a_i1*b_1j + a_i2*b_2j + ... + a_in*b_nj = Σ (from k=1 to n) a_ik * b_kj
  • Properties:
    • Non-commutative (Generally): AB ≠ BA in most cases. Sometimes AB is defined but BA is not, or vice versa. Even if both are defined, they may not be equal.
    • Associative Law: (AB)C = A(BC) (whenever both sides are defined).
    • Distributive Law:
      • A(B + C) = AB + AC
      • (A + B)C = AC + BC (whenever both sides are defined).
    • Existence of Multiplicative Identity: For every square matrix A, there exists an identity matrix I of the same order such that AI = IA = A.

5. Transpose of a Matrix

• Definition: If A = [a_ij] is an m × n matrix, then the transpose of A, denoted by A' or A^T, is the n × m matrix obtained by interchanging the rows and columns of A. A' = [a_ji]_(n x m).
  • Example: If A = [ 1 2 3 ], then A' = [ 1 4 ]
[ 4 5 6 ] [ 2 5 ]
[ 3 6 ]
• Properties of Transpose:
  • (A')' = A
  • (kA)' = kA' (where k is any constant)
  • (A + B)' = A' + B'
  • (AB)' = B'A' (Reversal Law) - Very Important!

6. Symmetric and Skew-Symmetric Matrices

• Symmetric Matrix: A square matrix A is symmetric if A' = A (i.e., a_ij = a_ji for all i, j).
  • Example: [ 1 2 3 ]
[ 2 5 -1 ]
[ 3 -1 0 ]
• Skew-Symmetric Matrix: A square matrix A is skew-symmetric if A' = -A (i.e., a_ij = -a_ji for all i, j).
  • Note: For i = j, a_ii = -a_ii, which implies 2a_ii = 0, so a_ii = 0. All diagonal elements of a skew-symmetric matrix must be zero.
  • Example: [ 0 2 -3 ]
[ -2 0 1 ]
[ 3 -1 0 ]
• Theorems (Important):
  • Theorem 1: For any square matrix A with real number entries, A + A' is a symmetric matrix and A - A' is a skew-symmetric matrix.
  • Theorem 2: Any square matrix A can be expressed uniquely as the sum of a symmetric matrix and a skew-symmetric matrix. A = (1/2)(A + A') + (1/2)(A - A'), where (1/2)(A + A') is symmetric and (1/2)(A - A') is skew-symmetric.

7. Elementary Operations (Transformations) of a Matrix

These are operations used to simplify matrices or find their inverse. There are six operations (three for rows, three for columns):

• (i) R_i ↔ R_j or C_i ↔ C_j (Interchange of any two rows or columns)
• (ii) R_i → kR_i or C_i → kC_i (Multiplication of the elements of any row or column by a non-zero number k)
• (iii) R_i → R_i + kR_j or C_i → C_i + kC_j (Adding to the elements of any row or column, the corresponding elements of another row or column multiplied by any non-zero number k)

8. Invertible Matrices

• Definition: If A is a square matrix of order m, and if there exists another square matrix B of the same order m such that AB = BA = I (where I is the identity matrix of order m), then A is said to be invertible, and B is called the inverse of A. The inverse is denoted by A⁻¹. So, AA⁻¹ = A⁻¹A = I.
• Uniqueness: If the inverse of a square matrix exists, it is unique.
• Condition for Invertibility (using Determinants - Chapter 4): A square matrix A is invertible if and only if its determinant is non-zero (det(A) ≠ 0). If det(A) = 0, the matrix is called a singular matrix. If det(A) ≠ 0, it's non-singular.
• Inverse using Elementary Operations: To find A⁻¹ using elementary row operations, write A = IA and apply a sequence of row operations on A on the LHS and I on the RHS until we get I = BA. The matrix B will be the inverse A⁻¹. (Similarly, for column operations, start with A = AI and apply column operations to get I = AB).
• Property: If A and B are invertible matrices of the same order, then (AB)⁻¹ = B⁻¹A⁻¹ (Reversal Law for Inverses).

Multiple Choice Questions (MCQs)

1. If a matrix has 12 elements, what are the possible orders it can have?
   a) 1x12, 2x6, 3x4
   b) 1x12, 12x1, 2x6, 6x2, 3x4, 4x3
   c) 1x12, 2x6, 3x4, 6x2, 12x1
   d) 2x6, 3x4, 4x3, 6x2

2. If A = [ a_ij ]_(2x3) such that a_ij = i + j, then the matrix A is:
   a) [ 2 3 4 ]
[ 3 4 5 ]
   b) [ 1 2 3 ]
[ 1 2 3 ]
   c) [ 2 3 ]
[ 3 4 ]
[ 4 5 ]
   d) [ 1 1 ]
[ 2 2 ]
[ 3 3 ]

3. If [ x+y 2 ] = [ 6 2 ], then the values of x and y are:
   [ 5+z xy] [ 5 8 ]
   a) x=2, y=4, z=0
   b) x=4, y=2, z=0
   c) x=3, y=3, z=0
   d) x=2, y=4, z=1

4. If A = [ 1 2 ] and B = [ 3 1 ], then A + B is:
   [ 3 4 ] [ 2 5 ]
   a) [ 4 3 ]
[ 5 9 ]
   b) [ 4 4 ]
[ 6 9 ]
   c) [ 3 2 ]
[ 6 20 ]
   d) Not possible

5. If A = [ 0 1 ] and B = [ 1 0 ], then AB is:
   [ 1 0 ] [ 0 1 ]
   a) [ 0 1 ]
[ 1 0 ]
   b) [ 1 0 ]
[ 0 1 ]
   c) [ 0 0 ]
[ 0 0 ]
   d) [ 1 1 ]
[ 1 1 ]

6. If A is a square matrix such that A^2 = A, then (I + A)^3 - 7A is equal to:
   a) A
   b) I - A
   c) I
   d) 3A

7. If A = [ cos(α) -sin(α) ], then A + A' equals:
   [ sin(α) cos(α) ]
   a) I
   b) [ 2cos(α) 0 ]
[ 0 2cos(α) ]
   c) O
   d) [ 0 -2sin(α) ]
[ 2sin(α) 0 ]

8. If A is a symmetric matrix, then A' is:
   a) Skew-symmetric
   b) Symmetric
   c) Zero matrix
   d) Identity matrix

9. If A is a square matrix, then A - A' is a:
   a) Symmetric matrix
   b) Skew-symmetric matrix
   c) Identity matrix
   d) Zero matrix

10. For two matrices A and B, the product AB is defined only if:
    a) Number of rows in A = Number of rows in B
    b) Number of columns in A = Number of columns in B
    c) Number of columns in A = Number of rows in B
    d) Number of rows in A = Number of columns in B

Answers to MCQs:

1. b) 1x12, 12x1, 2x6, 6x2, 3x4, 4x3
2. a) [ 2 3 4 ]
[ 3 4 5 ]
3. b) x=4, y=2, z=0 (Since x+y=6, xy=8 => x=4,y=2 or x=2,y=4. Also 5+z=5 => z=0)
4. a) [ 4 3 ]
[ 5 9 ]
5. a) [ 0 1 ]
[ 1 0 ]
6. c) I (Expand (I+A)^3 = I^3 + A^3 + 3I^2A + 3IA^2 = I + A^2A + 3A + 3A^2 = I + AA + 3A + 3A = I + A^2 + 6A = I + A + 6A = I + 7A. So, (I+A)^3 - 7A = I + 7A - 7A = I)
7. b) [ 2cos(α) 0 ]
[ 0 2cos(α) ] (A' = [ cos(α) sin(α) ]. A+A' = [ 2cos(α) 0 ])
   [ -sin(α) cos(α) ] [ 0 2cos(α) ]
8. b) Symmetric (If A is symmetric, A'=A. Then (A')' = A = A'. So A' is also symmetric)
9. b) Skew-symmetric matrix (This is Theorem 1)
10. c) Number of columns in A = Number of rows in B

Make sure you understand these concepts thoroughly. Practice problems involving matrix operations and properties, as they form the basis for many questions. Good luck!