Question 1 - Chapter 4 Class 12 Determinants (Important Question)
Last updated at Dec. 16, 2024 by Teachoo
Chapter 4 Class 12 Determinants
Ex 4.1, 7 (i)
Important
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Class 12
Important Questions for exams Class 12
- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
-
Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability
Transcript
Question 1 Without expanding the determinant, prove that |■8(a&a2&bc@b&b2&ca@c&c2&ab)| = |■8(1&a2&a3@1&b2&b3@1&c2&c3)| Solving L.H.S |■8(a&a2&bc@b&b2&ca@c&c2&ab)| Multiplying and dividing by abc = abc/abc |■8(a&a2&bc@b&b2&ca@c&c2&ab)| Multiplying a to R1, b to R2 & c to R3 = 1/abc |■8(a(𝑎)&𝑎(a2)&a(bc)@b(𝑏)&b(b2)&b (ca)@c(𝑐)&𝑐(c2)&c (ab))| Multiplying a to R1, b to R2 & c to R3 = 1/abc |■8(a(𝑎)&𝑎(a2)&a(bc)@b(𝑏)&b(b2)&b (ca)@c(𝑐)&𝑐(c2)&c (ab))| = 1/abc |■8(a2&a3&𝑎𝑏𝑐@b2&b3&𝑎𝑏𝑐@c2&c3&𝑎𝑏𝑐)| Taking abc common from C3 = 𝑎𝑏𝑐/𝑎𝑏𝑐 |■8(a2&a3&1@b2&b3&1@c2&c3&1)| = |■8(a2&a3&1@b2&b3&1@c2&c3&1)| Interchange C1 ↔ C3 = (–1) |■8(1&a3&a2@1&b3&b2@1&c3&c2)| Interchange C2 ↔ C3 = (–1) (–1) |■8(1&a2&a3@1&b2&b3@1&c2&c3)| = |■8(1&a2&a3@1&b2&b3@1&c2&c3)| = R.H.S. Hence Proved We know that If any two row or column of a determinant are interchanged, then sign of determinant changes.
