NCERT Class 12 Maths · Chapter 3 Matrices · Exercise 3.2 Question 3

✅ Step-by-Step Solution

1. For Product (1), the inner dimensions agree: 2. Its order is 2×2. Use cᵢⱼ=Σₖaᵢₖbₖⱼ.
2. Row 1: \(c_{11}=(a)(a) + (b)(b)=a^{2} + b^{2},\quad c_{12}=(a)(- b) + (b)(a)=0\).
3. Row 2: \(c_{21}=(- b)(a) + (a)(b)=0,\quad c_{22}=(- b)(- b) + (a)(a)=a^{2} + b^{2}\).
4. Product (1) = \(\left[\begin{matrix}a^{2} + b^{2} & 0\\0 & a^{2} + b^{2}\end{matrix}\right]\).
5. For Product (2), the inner dimensions agree: 1. Its order is 3×3. Use cᵢⱼ=Σₖaᵢₖbₖⱼ.
6. Row 1: \(c_{11}=(1)(2)=2,\quad c_{12}=(1)(3)=3,\quad c_{13}=(1)(4)=4\).
7. Row 2: \(c_{21}=(2)(2)=4,\quad c_{22}=(2)(3)=6,\quad c_{23}=(2)(4)=8\).
8. Row 3: \(c_{31}=(3)(2)=6,\quad c_{32}=(3)(3)=9,\quad c_{33}=(3)(4)=12\).
9. Product (2) = \(\left[\begin{matrix}2 & 3 & 4\\4 & 6 & 8\\6 & 9 & 12\end{matrix}\right]\).
10. For Product (3), the inner dimensions agree: 2. Its order is 2×3. Use cᵢⱼ=Σₖaᵢₖbₖⱼ.
11. Row 1: \(c_{11}=(1)(1) + (-2)(2)=-3,\quad c_{12}=(1)(2) + (-2)(3)=-4,\quad c_{13}=(1)(3) + (-2)(1)=1\).
12. Row 2: \(c_{21}=(2)(1) + (3)(2)=8,\quad c_{22}=(2)(2) + (3)(3)=13,\quad c_{23}=(2)(3) + (3)(1)=9\).
13. Product (3) = \(\left[\begin{matrix}-3 & -4 & 1\\8 & 13 & 9\end{matrix}\right]\).
14. For Product (4), the inner dimensions agree: 3. Its order is 3×3. Use cᵢⱼ=Σₖaᵢₖbₖⱼ.
15. Row 1: \(c_{11}=(2)(1) + (3)(0) + (4)(3)=14,\quad c_{12}=(2)(-3) + (3)(2) + (4)(0)=0,\quad c_{13}=(2)(5) + (3)(4) + (4)(5)=42\).
16. Row 2: \(c_{21}=(3)(1) + (4)(0) + (5)(3)=18,\quad c_{22}=(3)(-3) + (4)(2) + (5)(0)=-1,\quad c_{23}=(3)(5) + (4)(4) + (5)(5)=56\).
17. Row 3: \(c_{31}=(4)(1) + (5)(0) + (6)(3)=22,\quad c_{32}=(4)(-3) + (5)(2) + (6)(0)=-2,\quad c_{33}=(4)(5) + (5)(4) + (6)(5)=70\).
18. Product (4) = \(\left[\begin{matrix}14 & 0 & 42\\18 & -1 & 56\\22 & -2 & 70\end{matrix}\right]\).
19. For Product (5), the inner dimensions agree: 2. Its order is 3×3. Use cᵢⱼ=Σₖaᵢₖbₖⱼ.
20. Row 1: \(c_{11}=(2)(1) + (1)(-1)=1,\quad c_{12}=(2)(0) + (1)(2)=2,\quad c_{13}=(2)(1) + (1)(1)=3\).
21. Row 2: \(c_{21}=(3)(1) + (2)(-1)=1,\quad c_{22}=(3)(0) + (2)(2)=4,\quad c_{23}=(3)(1) + (2)(1)=5\).
22. Row 3: \(c_{31}=(-1)(1) + (1)(-1)=-2,\quad c_{32}=(-1)(0) + (1)(2)=2,\quad c_{33}=(-1)(1) + (1)(1)=0\).
23. Product (5) = \(\left[\begin{matrix}1 & 2 & 3\\1 & 4 & 5\\-2 & 2 & 0\end{matrix}\right]\).
24. For Product (6), the inner dimensions agree: 3. Its order is 2×2. Use cᵢⱼ=Σₖaᵢₖbₖⱼ.
25. Row 1: \(c_{11}=(3)(2) + (-1)(1) + (3)(3)=14,\quad c_{12}=(3)(-3) + (-1)(0) + (3)(1)=-6\).
26. Row 2: \(c_{21}=(-1)(2) + (0)(1) + (2)(3)=4,\quad c_{22}=(-1)(-3) + (0)(0) + (2)(1)=5\).
27. Product (6) = \(\left[\begin{matrix}14 & -6\\4 & 5\end{matrix}\right]\).