Matrices Class 12 Notes

• Row and column matrices have one row or one column; a square matrix has m = n.
• Diagonal matrix: all off-diagonal entries zero; identity I: diagonal entries 1.
• Scalar and zero matrices: all diagonal entries equal (scalar) or every entry zero.
• A symmetric matrix satisfies Aᵀ = A; a skew-symmetric matrix satisfies Aᵀ = −A. The main diagonal of a skew-symmetric matrix is always zero.

• Two matrices add only if they have the same order:
• $(A+B{)}_{ij}=a_{ij}+b_{ij}$
• .
• Scalar multiplication:
• $(kA{)}_{ij}=k{a}_{ij}$
• .
• Addition is commutative and associative; it distributes over scalar multiplication.
• The transpose of a sum:
• $(A+B{)}^T=A^T+B^T$
• .

• Matrix multiplication is NOT commutative: AB ≠ BA in general.
• It is associative and distributive over addition.
• A is orthogonal to being invertible separately:
• $(AB{)}^{-1}=B^{-1}{A}^{-1}$
• .
• Transpose of a product:
• $(AB{)}^T=B^T{A}^T$
• .

• A square matrix A is invertible (non-singular) iff
• $|A|\ne 0$
• .
• Every square matrix is the sum of a symmetric and a skew-symmetric matrix:
• $A=\frac{A+A^T}{2}+\frac{A-A^T}{2}$
• .
• For an invertible matrix,
• $A{A}^{-1}=A^{-1}A=I$
• .

Example: For $A=\left(\begin{array}{cc}2& 3\\ 1& 4\end{array}\right)$, find |A| and A⁻¹.

Solution: $|A|=2\cdot 4-3\cdot 1=5\ne 0$, so A is invertible. $\text{adj}(A)=\left(\begin{array}{cc}4& -3\\ -1& 2\end{array}\right)$, and $A^{-1}=\frac{1}{5}\left(\begin{array}{cc}4& -3\\ -1& 2\end{array}\right)$.

Exercises (NCERT Q1–Q61)

• 1) In the matrix , write the order, number of elements, and the elements
• 2) If a matrix has 24 elements, what are the possible orders? What if it has 13 elements?
• 3) If a matrix has 18 elements, what are the possible orders? What if it has 5 elements?
• 4) Construct a 2 × 2 matrix , where
• 5) Construct a 3 × 4 matrix , where
• 6) Find x and y from the equation
• 7) Find the values of a, b, c, d from
• 8) Find a and b from
• 9) Find x, y, z, w from
• 10) Find the values of a, b, c, d from the equation
• 11) Let , , . Find each of the following: (i) A + B, (ii) A − B, (iii) 3A − C, (iv) 3A + 4B.
• 12) Compute the following products: (i) ; (ii) ; (iii) ; (iv)
• 13) Let and . Verify that (i) A + B = B + A, (ii) A − B ≠ B − A.
• 14) Find A² − B² if and
• 15) Compute for the matrices in Question 3. Is it equal to ?
• 16) Solve for X and Y: and
• 17) Find X if
• 18) If , show that
• 19) If , prove that
• 20) If and , verify that (i) (ii)
• 21) If , , , verify that
• 22) Verify associativity of matrix multiplication: , , . Show that (AB)C = A(BC).
• 23) Given and , find AB and BA. Are they equal?
• 24) Let . Verify that
• 25) Let . Show that for all n ∈ ℕ.
• 26) Find the inverse of using the fact that
• 27) If show that . Hence find A⁻¹.
• 28) If , , find (AB)⁻¹ and verify it equals B⁻¹A⁻¹.
• 29) If , show that and hence find A⁻¹.
• 30) If and , verify that (i) (ii)
• 31) If , find and show that A² ≠ I in general.
• 32) If , verify that
• 33) If , show that and hence find A⁻¹.
• 34) If A and B are symmetric matrices of the same order and AB = BA, prove that AB is symmetric.
• 35) Let and . Find (i) (A+B)² (ii) A² + B² + 2AB. Are they equal?
• 36) Show that there exist matrices A and B such that AB ≠ BA.
• 37) If and , verify that
• 38) If , prove that (zero matrix).
• 39) Find the transpose of each of the following matrices: (i) (ii) (iii)
• 40) If and , verify that (i) (ii)
• 41) If and , verify that (BA)′ = A′B′.
• 42) If , find and show it is a symmetric matrix.
• 43) Find the values of x and y so that is symmetric.
• 44) Find the values of x and y so that is symmetric.
• 45) Find the values of a and b so that is skew-symmetric.
• 46) Find the values of a and b so that is skew-symmetric.
• 47) Express as the sum of a symmetric and a skew-symmetric matrix.
• 48) Express as the sum of a symmetric and a skew-symmetric matrix.
• 49) If A and B are symmetric matrices of the same order, show that AB is symmetric if and only if AB = BA.
• 50) Show that every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.
• 51) If A is a square matrix such that A is both symmetric and skew-symmetric, prove that A is a zero matrix.
• 52) If A and B are square matrices of the same order, show that (i) is symmetric (ii) is skew-symmetric.
• 53) If , show that A is symmetric if and only if it is a scalar multiple of the identity matrix.
• 54) Show that for a square matrix A, A + A′ is always symmetric.
• 55) Show that the matrix is skew-symmetric for all values of a, b, c.
• 56) Show that the matrix satisfies
• 57) If A and B are symmetric matrices of the same order, prove that AB − BA is skew-symmetric.
• 58) State the three types of elementary row operations that can be performed on a matrix. Give an example of each.
• 59) Define elementary column operations on a matrix. How are they related to elementary row operations?
• 60) Using elementary row operations, find the inverse of
• 61) Using elementary row operations, find the inverse of

Answers at a glance

1. 1
   Q1 → Order: 3 × 4; elements: 12; a₁₃ = 19, a₂₁ = 35, a₃₃ = −5, a₂₄ = 12, a₂₃ = 5/3.
2. 2
   Q2 → 24 elements: orders 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1 (8 orders). 13 elements: orders 1×13, 13×1.
3. 3
   Q3 → 18 elements: orders 1×18, 2×9, 3×6, 6×3, 9×2, 18×1. 5 elements: orders 1×5, 5×1.
4. 4
   Q4 → answer not in source — verify against the textbook
5. 5
   Q5 → answer not in source — verify against the textbook
6. 6
   Q6 → x = 2, y = 4.
7. 7
   Q7 → a + b = 6 and c + d = 8 (infinitely many solutions, e.g. a=2, b=4, c=3, d=5).
8. 8
   Q8 → a = 4/5, b = 7/5.
9. 9
   Q9 → x = −1, y = 4, z = 2, w = 2.
10. 10
    Q10 → a = 1, b = 2, c = −3, d = 22.
11. 11
    Q11 → answer not in source — verify against the textbook
12. 12
    Q12 → answer not in source — verify against the textbook
13. 13
    Q13 → answer not in source — verify against the textbook
14. 14
    Q14 → answer not in source — verify against the textbook
15. 15
    Q15 → (A+B)(A−B) = [−10,10; −10,10]. This is NOT equal to A² − B², so the identity (A+B)(A−B) = A² − B² fails for matrix multiplication.
16. 16
    Q16 → answer not in source — verify against the textbook
17. 17
    Q17 → answer not in source — verify against the textbook
18. 18
    Q18 → A² − 5A + 7I = O, verified.
19. 19
    Q19 → A³ − 6A² + 7A + 2I = O, verified.
20. 20
    Q20 → Both identities verified: (AB)′ = B′A′ and (2A − B)′ = 2A′ − B′ hold.
21. 21
    Q21 → A(B+C) = AB + AC = [8,15; 20,37]. Verified.
22. 22
    Q22 → answer not in source — verify against the textbook
23. 23
    Q23 → AB = BA = O. They are equal.
24. 24
    Q24 → (A²)²⁰ = A⁴⁰ = I₂, verified.
25. 25
    Q25 → answer not in source — verify against the textbook
26. 26
    Q26 → answer not in source — verify against the textbook
27. 27
    Q27 → answer not in source — verify against the textbook
28. 28
    Q28 → answer not in source — verify against the textbook
29. 29
    Q29 → answer not in source — verify against the textbook
30. 30
    Q30 → (A+B)′ = A′ + B′ and (AB)′ = B′A′, both verified for these matrices.
31. 31
    Q31 → answer not in source — verify against the textbook
32. 32
    Q32 → (A⁻¹)⁻¹ = A, verified.
33. 33
    Q33 → answer not in source — verify against the textbook
34. 34
    Q34 → (AB)′ = B′A′ = BA = AB, so AB is symmetric.
35. 35
    Q35 → (A+B)² = 2I = [2,0; 0,2]. A²+B²+2AB = [2,2; −2,2]. They are NOT equal because AB ≠ BA, so (A+B)² = A²+AB+BA+B² ≠ A²+2AB+B².
36. 36
    Q36 → A = [0,1; 0,0], B = [1,0; 0,0]: AB = O ≠ BA = [0,1; 0,0].
37. 37
    Q37 → (AB)′ = B′A′ = [11,7; 11,7]. Verified.
38. 38
    Q38 → A² = O. This is a nonzero matrix whose square is zero, illustrating that nonzero matrices can be nilpotent of index 2.
39. 39
    Q39 → answer not in source — verify against the textbook
40. 40
    Q40 → Both identities verified by direct computation.
41. 41
    Q41 → (BA)′ = A′B′, verified by direct computation.
42. 42
    Q42 → answer not in source — verify against the textbook
43. 43
    Q43 → A is symmetric for all real x and y — the off-diagonal entries are both x, so A′ = A automatically.
44. 44
    Q44 → x = −3; y can be any real number.
45. 45
    Q45 → No solution — the (2,2) entry is 4 ≠ 0, but every diagonal entry of a skew-symmetric matrix must be 0.
46. 46
    Q46 → No solution — the (2,2) entry is 1 ≠ 0 (diagonal of a skew-symmetric matrix must be 0).
47. 47
    Q47 → answer not in source — verify against the textbook
48. 48
    Q48 → A is already symmetric (A′ = A), so the symmetric part is A itself and the skew-symmetric part is the zero matrix O. Hence A = A + O.
49. 49
    Q49 → (AB)′ = B′A′ = BA (using A′=A, B′=B). AB symmetric ⟺ (AB)′ = AB ⟺ BA = AB.
50. 50
    Q50 → A = (A+A′)/2 + (A−A′)/2 gives the unique decomposition into symmetric + skew-symmetric parts.
51. 51
    Q51 → A = A′ = −A ⟹ 2A = O ⟹ A = O.
52. 52
    Q52 → (i) AB′ + BA′ is symmetric since (AB′ + BA′)′ = BA′ + AB′. (ii) AB′ − BA′ is skew-symmetric since (AB′ − BA′)′ = BA′ − AB′ = −(AB′ − BA′).
53. 53
    Q53 → A symmetric requires sin θ = 0, giving θ = nπ and A = ±I (a scalar multiple of the identity).
54. 54
    Q54 → (A + A′)′ = A′ + A = A + A′, so A + A′ is always symmetric.
55. 55
    Q55 → B is skew-symmetric: B′ = −B for all a, b, c. The diagonal is all zeros and off-diagonal pairs satisfy b′ᵢⱼ = −b′ⱼᵢ.
56. 56
    Q56 → A² = [2,−2; −2,2] = 2A, verified.
57. 57
    Q57 → (AB − BA)′ = B′A′ − A′B′ = BA − AB = −(AB − BA). Skew-symmetric.
58. 58
    Q58 → The three elementary row operations are: (1) Interchange of two rows (Rᵢ ↔ Rⱼ); (2) Multiplication of a row by a non-zero scalar (Rᵢ → kRᵢ, k ≠ 0); (3) Addition of a scalar multiple of one row to another (Rᵢ → Rᵢ + kRⱼ).
59. 59
    Q59 → Elementary column operations mirror row operations applied to columns: (1) Cᵢ ↔ Cⱼ, (2) Cᵢ → kCᵢ (k≠0), (3) Cᵢ → Cᵢ + kCⱼ. They correspond to right-multiplication by elementary matrices (versus left-multiplication for row operations).
60. 60
    Q60 → answer not in source — verify against the textbook
61. 61
    Q61 → answer not in source — verify against the textbook

MCQ Questions

1 Mark Questions

1. 1
   State the condition under which AB = BA holds for two matrices.
2. 2
   What is the order of the zero matrix that acts as the additive identity for 2 × 3 matrices?
3. 3
   When is a square matrix said to be a diagonal matrix?
4. 4
   Write the number of elements in a matrix of order 3 × 3. (Ans: 9)

2 Mark Questions

1. 1
   If A = [2 1; 0 3] and B = [1 2; 1 0], find AB and BA and check whether they are equal.
2. 2
   If A = [x 1; 0 2] and A satisfies A² = [5 3; 0 4], find x. (Ans: x = 1)
3. 3
   Show that the sum of a matrix and its transpose is symmetric for A = [1 2; 3 4]. (Ans: A + Aᵀ = [2 5; 5 8])

3 Mark Questions

1. 1
   Express the matrix A = [3 5; 1 −1] as the sum of a symmetric and a skew-symmetric matrix. (Ans: (A + Aᵀ)/2 + (A − Aᵀ)/2)
2. 2
   Prove that the product of two non-singular matrices is non-singular, and give the determinant relation. (Ans: |AB| = |A||B|)
3. 3
   If A = [1 1; 0 1], find Aⁿ for a positive integer n by induction. (Ans: Aⁿ = [1 n; 0 1])

4 Mark Questions

1. 1
   Using elementary row operations, reduce the matrix A = [2 1; 3 4] to the identity matrix and hence (or otherwise) show A⁻¹ = (1/5)[4 −1; −3 2].
2. 2
   State the conditions for the product AB to be defined. Using A = [1 3; 2 4] and B = [5 0; 2 1], verify that (Aᵀ)ᵀ = A, (AB)ᵀ = BᵀAᵀ, and (AB)⁻¹ = B⁻¹A⁻¹.