# Ex 4.2, 5

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 4.2, 5 Using the property of determinants and without expanding, prove that: b+cq+ry+zc+ar+pz+xa+bp+qx+y = 2 apxbqycrz Taking L.H.S b+cq+ry+zc+ar+pz+xa+bp+qx+y Applying R3 → R3 + R2 + R1 = b+𝑐𝑞+𝑟𝑦+𝑧c+ar+𝑝z+xa+b+c+a+b+c𝑝+𝑞+𝑟+𝑞+𝑞+𝑟x+𝑦+𝑧+𝑥+𝑦+𝑧 = b+𝑐𝑞+𝑟𝑦+𝑧c+ar+𝑝z+x𝟐(a+b+c)𝟐(𝑝+𝑞+𝑟)𝟐(x+𝑦+𝑧) Taking 2 common from R3 = 2 b+𝑐𝑞+𝑟𝑦+𝑧c+ar+𝑝z+x(a+b+c)(𝑝+𝑞+𝑟)(x+𝑦+𝑧) Applying R1 → R1 – R3 = 2 b+𝑐−𝑎−𝑏−𝑐𝑞+𝑟−𝑝−𝑞−𝑟𝑦+𝑧−𝑥−𝑦−𝑧c+ar+𝑝z+xa+b+c𝑝+𝑞+𝑟x+𝑦+𝑧 = 2 −𝑎−𝑝−𝑥c+ar+𝑝z+xa+b+c𝑝+𝑞+𝑟x+𝑦+𝑧 Applying R2 → R2 – R3 = 2 −𝑎−𝑝−𝑥c+a−a−b−cr+𝑝−𝑝−𝑞−𝑟z+x−x−y−za+b+c𝑝+𝑞+𝑟x+𝑦+𝑧 = 2 −𝑎−𝑝−𝑥−b−𝑞−ya+b+c𝑝+𝑞+𝑟x+𝑦+𝑧 Applying R3 → R3 + R1 + R2 = 2 −𝑎−𝑝−𝑥−b−𝑞−ya+b+c−𝑎−𝑏𝑝+𝑞+𝑟−𝑝−𝑞x+𝑦+𝑧−𝑥−𝑦 = 2 −𝑎−𝑝−𝑥−b−𝑞−y𝑐𝑟𝑧 Taking –1 Common from R1 & R3 = 2 ( –1) ( –1) 𝑎𝑝𝑥b𝑞y𝑐𝑟𝑧 = 2 𝑎𝑝𝑥b𝑞y𝑐𝑟𝑧 = R.H.S Hence Proved

Chapter 4 Class 12 Determinants

Serial order wise

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CA Maninder Singh

CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .