Class 12 Mathematics Notes Chapter 3 (Matrices) – Examplar Problems (English) Book

Alright class, let's get started with Chapter 3: Matrices. This is a fundamental chapter, not just for your board exams but also forms the basis for many concepts in higher mathematics and various competitive government exams. We'll focus on the core concepts, properties, and problem types, keeping the NCERT Exemplar perspective in mind, which often tests deeper understanding.

Chapter 3: Matrices - Detailed Notes for Exam Preparation

1. Definition and Notation:

• A matrix is an ordered rectangular array of numbers or functions. These numbers or functions are called the elements or entries of the matrix.
• Matrices are usually denoted by capital letters (e.g., A, B, C).
• The elements are enclosed in square brackets [] or parentheses ().
• 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').
• Element Notation: The element in the i-th row and j-th column of a matrix A is denoted by a_ij or (A)_ij.
  So, A = [a_ij]_{m×n}.
• Example:
  A = [[1, -2, 5], [0, sqrt(3), 7]] is a matrix of order 2 × 3. Here, a_11 = 1, a_12 = -2, a_23 = 7, etc.
• The total number of elements in an m × n matrix is mn.

2. Types of Matrices:

• (i) Column Matrix: A matrix with only one column (order m × 1).
  • Example: B = [[1], [-3], [0]] (Order 3 × 1)
• (ii) Row Matrix: A matrix with only one row (order 1 × n).
  • Example: C = [2, 0, -5] (Order 1 × 3)
• (iii) Square Matrix: A matrix where the number of rows equals the number of columns (m = n). It's called a square matrix of order n.
  • Example: D = [[1, 2], [3, 4]] (Order 2)
  • The elements a_11, a_22, ..., a_nn form 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: E = [[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: F = [[3, 0, 0], [0, 3, 0], [0, 0, 3]]
• (vi) Identity Matrix (or Unit Matrix): A scalar matrix where all diagonal elements are 1 (i.e., k=1). Denoted by I or I_n (for order n). It acts as the multiplicative identity for matrix multiplication.
  • Example: I_2 = [[1, 0], [0, 1]], I_3 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
• (vii) Zero Matrix (or Null Matrix): A matrix where all elements are zero. Denoted by O. The order is usually clear from the context.
  • Example: O = [[0, 0], [0, 0]]

3. Equality of Matrices:

• Two matrices A = [a_ij] and B = [b_ij] are equal if:
  • (i) They are of the same order.
  • (ii) Each corresponding element is equal (a_ij = b_ij for all i and j).
• This property is often used to solve for unknown values within matrices.

4. Operations on Matrices:

• (i) Addition of Matrices:
  • Condition: Only matrices of the same order can be added.
  • Process: If A = [a_ij]_{m×n} and B = [b_ij]_{m×n}, then A + B = [a_ij + b_ij]_{m×n}. (Add corresponding elements).
  • Properties:
    • Commutative: A + B = B + A
    • Associative: (A + B) + C = A + (B + C)
    • Additive Identity: A + O = O + A = A (where O is the zero matrix of the same order as A)
    • 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) Subtraction of Matrices:
  • Condition: Same order.
  • Process: A - B = A + (-B). (Subtract corresponding elements).
• (iii) Multiplication of a Matrix by a Scalar:
  • Process: If k is a scalar and A = [a_ij]_{m×n}, then kA = [k * a_ij]_{m×n}. (Multiply every element by k).
  • Properties:
    • k(A + B) = kA + kB
    • (k + l)A = kA + lA (where k, l are scalars)
• (iv) Multiplication of Matrices:
  • Condition: The number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B). If A is m × n and B is n × p, then the product AB is defined and its order is m × p.
  • Process: If C = AB, then the element c_ij (element in i-th row, j-th column of C) 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 summing the products.
    c_ij = sum_{k=1}^{n} (a_{ik} * b_{kj})
  • Example: If A = [[1, 2], [3, 4]] (2x2) and B = [[5, 6], [7, 8]] (2x2), then AB is (2x2).
    AB = [[(1*5 + 2*7), (1*6 + 2*8)], [(3*5 + 4*7), (3*6 + 4*8)]] = [[19, 22], [43, 50]]
  • Properties:
    • Not Commutative (Generally): AB ≠ BA in most cases. Sometimes BA may not even be defined when AB is.
    • Associative: (AB)C = A(BC) (if orders are compatible for multiplication).
    • Distributive: A(B + C) = AB + AC and (A + B)C = AC + BC (if orders are compatible).
    • Multiplicative Identity: AI = IA = A (where I is the identity matrix of appropriate order).
    • If AB = O, it does not necessarily imply A = O or B = O.

5. Transpose of a Matrix:

• The transpose of a matrix A = [a_ij]_{m×n}, denoted by A' or A^T, is the matrix obtained by interchanging its rows and columns. Its order is n × m.
  A' = [a_ji]_{n×m}.
• Example: If A = [[1, 2, 3], [4, 5, 6]], then A' = [[1, 4], [2, 5], [3, 6]].
• Properties of Transpose:
  • (A')' = A
  • (kA)' = kA' (where k is a scalar)
  • (A + B)' = A' + B'
  • (Reversal Law) (AB)' = B'A' (Very Important!)

6. Symmetric and Skew-Symmetric Matrices:

• Condition: Must be a square matrix.
• (i) 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, 4, 5], [3, 5, 6]]
• (ii) 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, 1, -2], [-1, 0, 3], [2, -3, 0]]
• Theorems (Important for Exemplar/Exams):
  • For any square matrix A, A + A' is a symmetric matrix.
  • For any square matrix A, A - A' is a skew-symmetric matrix.
  • Any square matrix A can be expressed uniquely as the sum of a symmetric matrix (P) and a skew-symmetric matrix (Q).
    A = P + Q, where P = 1/2 (A + A') and Q = 1/2 (A - A').

7. Elementary Operations (Transformations) of a Matrix:

• These are fundamental operations used to find the inverse of a matrix and solve systems of linear equations. There are six operations (3 for rows, 3 for columns):
  • (i) Interchange: Interchange of any two rows (R_i ↔ R_j) or two columns (C_i ↔ C_j).
  • (ii) Multiplication by Scalar: Multiplication of the elements of any row (R_i → kR_i) or column (C_i → kC_i) by a non-zero scalar k.
  • (iii) Addition: Addition to the elements of any row (R_i → R_i + kR_j) or column (C_i → C_i + kC_j) the corresponding elements of another row/column multiplied by a non-zero scalar k.
• Equivalent Matrices: Two matrices are equivalent if one can be obtained from the other by a sequence of elementary operations.

8. Invertible Matrices (and Inverse by Elementary Operations):

• Condition: Only square matrices can be invertible.
• Definition: A square matrix A of order n is invertible if there exists another square matrix B of the same order n such that AB = BA = I_n. Matrix B is called the inverse of A, denoted by A⁻¹.
• Uniqueness: If a square matrix has an inverse, it is unique.
• Finding Inverse using Elementary Row Operations:
  1. Write A = IA (where I is the identity matrix of the same order as A).
  2. Apply a sequence of elementary row operations to the equation A = IA until you get I = BA.
  3. The matrix B on the right-hand side is the inverse A⁻¹.
  • Important: Apply the same row operations simultaneously to A (on the LHS) and I (the pre-factor of A on the RHS).
• Finding Inverse using Elementary Column Operations:
  1. Write A = AI.
  2. Apply a sequence of elementary column operations to the equation A = AI until you get I = AB.
  3. The matrix B on the right-hand side is the inverse A⁻¹.
  • Important: Apply the same column operations simultaneously to A (on the LHS) and I (the post-factor of A on the RHS).
• Note: If, during the application of elementary operations, we obtain a row or column consisting entirely of zeros on the LHS matrix, then A⁻¹ does not exist (the matrix is singular).
• 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) 12x1, 6x2, 4x3
   (D) 1x12, 2x6, 3x4, 4x3, 6x2

2. Construct a 2x2 matrix A = [a_ij] whose elements are given by a_ij = (i + 2j)² / 2.
   (A) [[9/2, 25/2], [8, 18]]
   (B) [[9/2, 25/2], [4, 9]]
   (C) [[9/2, 25/2], [8, 36]]
   (D) [[4.5, 12.5], [8, 18]]

3. If [[x+y, 2], [5+z, xy]] = [[6, 2], [5, 8]], find the values of x, y, z.
   (A) x=2, y=4, z=0
   (B) x=4, y=2, z=0
   (C) x=3, y=3, z=0
   (D) Either (A) or (B)

4. If A and B are square matrices of the same order such that AB = A and BA = B, then A² + B² is equal to:
   (A) AB
   (B) A + B
   (C) 2AB
   (D) I

5. If A = [[cos α, -sin α], [sin α, cos α]], then A + A' equals:
   (A) [[2cos α, 0], [0, 2cos α]]
   (B) [[0, 0], [0, 0]]
   (C) [[2cos α, -2sin α], [2sin α, 2cos α]]
   (D) I

6. If A and B are symmetric matrices of the same order, then (AB' - BA')' is:
   (A) Symmetric matrix
   (B) Skew-symmetric matrix
   (C) Null matrix
   (D) Identity matrix

7. If A = [[3, 5], [7, 9]] is written as A = P + Q, where P is symmetric and Q is skew-symmetric, then the matrix P is:
   (A) [[3, 6], [6, 9]]
   (B) [[0, -1], [1, 0]]
   (C) [[3, 5], [7, 9]]
   (D) [[6, 12], [12, 18]]

8. If A is a matrix of order m × n and B is a matrix such that AB' and B'A are both defined, then the order of matrix B is:
   (A) m × n
   (B) n × m
   (C) n × n
   (D) m × m

9. A square matrix A is invertible if and only if:
   (A) It is a diagonal matrix
   (B) It is a symmetric matrix
   (C) It is non-singular (determinant is non-zero - concept from next chapter, but essential)
   (D) Applying elementary row operations leads to an identity matrix.

10. If A and B are invertible matrices of the same order, then (AB)⁻¹ is equal to:
    (A) A⁻¹B⁻¹
    (B) B⁻¹A⁻¹
    (C) AB
    (D) BA

Answers to MCQs:

1. (B) - Factors of 12 are 1, 12, 2, 6, 3, 4. Possible orders are m x n where mn=12.
2. (D) - a_11 = (1+2*1)²/2 = 9/2 = 4.5, a_12 = (1+2*2)²/2 = 25/2 = 12.5, a_21 = (2+2*1)²/2 = 16/2 = 8, a_22 = (2+2*2)²/2 = 36/2 = 18.
3. (D) - x+y=6, xy=8, 5+z=5 => z=0. Solving x+y=6 and xy=8 gives x(6-x)=8 => x²-6x+8=0 => (x-2)(x-4)=0. So x=2, y=4 or x=4, y=2.
4. (B) - A² = A*A = A(BA) = (AB)A = AA = A. Similarly B² = B*B = B(AB) = (BA)B = BB = B. So A² + B² = A + B.
5. (A) - A' = [[cos α, sin α], [-sin α, cos α]]. A + A' = [[cos α + cos α, -sin α + sin α], [sin α - sin α, cos α + cos α]] = [[2cos α, 0], [0, 2cos α]].
6. (B) - Given A'=A, B'=B. Let C = AB' - BA' = AB - BA. Then C' = (AB - BA)' = (AB)' - (BA)' = B'A' - A'B' = BA - AB = -(AB - BA) = -C. Since C' = -C, it is skew-symmetric.
7. (A) - P = 1/2 (A + A'). A' = [[3, 7], [5, 9]]. A + A' = [[6, 12], [12, 18]]. P = 1/2 [[6, 12], [12, 18]] = [[3, 6], [6, 9]].
8. (D) - A is m × n. Let B be p × q. Then B' is q × p. AB' is defined means n = q. B'A is defined means p = m. So B is m × n. (Correction: Let B be p x q. B' is q x p. AB' is (m x n)(q x p). Defined means n=q. Result is m x p. B'A is (q x p)(m x n). Defined means p=m. So B is m x n. Let's recheck the question options carefully. Ah, the question asks for the order of B. We found B is p x q where p=m and q=n. So B is m x n. Let's re-verify the logic. A is m x n. B is p x q. B' is q x p. AB' requires n=q. B'A requires p=m. So B is m x n. Option (A) is m x n. Let me re-read the options and my derivation. Okay, p=m and q=n. So the order of B is m x n. Option (A) is correct. Let me double check the options again. (A) m x n (B) n x m (C) n x n (D) m x m. My derivation leads to m x n. Let me re-evaluate the question based on the provided answer D. If B is m x m, then B' is m x m. AB' is (m x n)(m x m). This is defined only if n=m. B'A is (m x m)(m x n). This is defined only if m=m. So if B is m x m, A must be m x m. This seems too restrictive. Let's re-evaluate the derivation. A is m x n. B is p x q. B' is q x p. AB' defined => n=q. B'A defined => p=m. So B is m x n. Why would the answer be (D)? Let's assume B is m x m (order m). B' is m x m. AB' is (m x n)(m x m). Defined if n=m. B'A is (m x m)(m x n). Defined if m=m. So if B is m x m, then A must be m x m. Let's assume B is n x n. B' is n x n. AB' is (m x n)(n x n). Defined if n=n. B'A is (n x n)(m x n). Defined if n=m. So if B is n x n, A must be n x n. Let's assume B is n x m. B' is m x n. AB' is (m x n)(m x n). Defined if n=m. B'A is (m x n)(m x n). Defined if n=m. So if B is n x m, A must be m x m. Let's assume B is m x n. B' is n x m. AB' is (m x n)(n x m). Defined if n=n. B'A is (n x m)(m x n). Defined if m=m. So B is m x n works for any A of order m x n. There seems to be an issue with the provided options or the question itself, as m x n (Option A) is the most general correct answer based on the conditions. However, competitive exams sometimes have tricky questions. Let's reconsider if there's an interpretation favoring (D) m x m. If AB' and B'A are both defined, maybe there's an implicit assumption they result in matrices that can be compared or used further, often implying square matrices? If AB' results in m x p = m x m and B'A results in q x n = n x n. This requires p=m and q=n. From earlier, n=q and p=m. So this implies B is m x n. This still leads to A. Let's stick with the derivation: B must be of order m x n. Option (A). Self-correction: Re-reading standard problems, often the context implies operations that lead back to square matrices or allow further operations. Let's re-examine the condition AB' and B'A are defined. A(m x n), B(p x q), B'(q x p). AB' defined => n=q. Product is m x p. B'A defined => p=m. Product is q x n. So B is m x n. The products are m x m and n x n. If we need these products to be potentially comparable or used in subsequent steps common in linear algebra (like addition, equality), often m=n is implied, making A and B square. But the question only states they are defined. The most direct answer is m x n. I'll stick with (A) based purely on the definition criteria. However, if forced to choose from potentially flawed options common in some exams, and knowing m x m or n x n are common requirements for further matrix properties, (D) or (C) might be intended answers in a specific context, but based only on the definition condition, (A) is correct. Let's assume the question intended for A and B to be square matrices for simplicity, which is common. If A is square (n x n), then B must be n x n. If A is m x n, B must be m x n. Let's assume option (D) m x m is the intended answer, perhaps implying A is also square m x m (so n=m). Let's proceed with (A) as derived, but acknowledge potential ambiguity in exam questions. Let's select (A) based on derivation.
9. (D) - While non-singular (C) is the condition checked using determinants (Chapter 4), the definition in Chapter 3 context relates to the existence of B such that AB=BA=I, which is found/verified using elementary operations. If elementary row operations can transform A to I, then A is invertible.
10. (B) - This is the standard reversal law for inverses.

Final check on Q8: The most general condition derived is B is m x n. This is option (A). Let's assume (A) is the correct answer.

Revised Answers:

1. (B)
2. (D)
3. (D)
4. (B)
5. (A)
6. (B)
7. (A)
8. (A)
9. (D) (In the context of Chapter 3 methods; (C) is also correct but uses concepts from Chapter 4)
10. (B)

Make sure you understand the definitions, properties, and especially the conditions under which operations are defined. Practice problems involving matrix multiplication, transpose properties, symmetric/skew-symmetric decomposition, and finding inverses using elementary operations, as these are frequently tested. Good luck!