# Ex 4.2, 14 - Chapter 4 Class 12 Determinants

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Ex 4.2, 14 By using properties of determinants, show that: a2+1 ab ac ab b2+1 bc ca cb c2+1 = 1 + a2 + b2 + c2 Taking L.H.S a2+1 ab ac ab b2+1 bc ca cb c2+1 Multiplying & Dividing by abc = a2+1 ab ac ab b2+1 bc ca cb c2+1 Multiplying 1st row by a, 2nd row by b & 3rd row by c ( R1 aR1 , R2 bR3 , R3 bR3 ) = 1 (a2+1) (ab) (ac) (ab) (b2+1) (bc) (ca) (cb) (c2+1) = 1 a3+a 2b 2c ab2 b3+b 2c c2a 2b c3+c Applying R1 R1 + R2 + R3 = 1 a3+a+ 2+ 2 2b+b3+b+c2b 2c+b2c+c3+c ab2 b3+b 2c c2a 2b c3+c = 1 a( + + + ) ( + + + ) ( + + +1) ab2 b3+b 2c c2a 2b c3+c Taking (1+ 2+ 2+ 2) common from 1st Row = ( + + + ) a ab2 b3+b 2c c2a 2b c(c3+1) Taking a common from C1 ,b from C2 & c from C3 . = (1+ 2+ 2+ 2) 1 1 1 b2 b3+1 2 c2 2 c2+1 Applying C1 C1 C2 = (1+ 2+ 2+ 2) 1 1 b2 2 1 b2+1 2 c2 c2 2 c2+1 = (1+ 2+ 2+ 2) 1 1 1 b2+1 2 0 2 c2+1 Applying C2 C2 C3 = (1+ 2+ 2+ 2) 0 1 1 b2+1 2 2 0 2 2 1 c2+1 = (1+ 2+ 2+ 2) 0 1 1 1 2 0 1 c2+1 Expanding along R1 = (1+ 2+ 2+ 2) 1 1 2 1 2+1 0 1 2 0 2+1 + 0 1 1 0 1 = (1 + a2 + b1 + c2) (0 0 + 1) = (1 + a2 + b1 + c2) (1) = (1 + a2 + b1 + c2) = R.H.S = Hence proved

Using Property 5 (Determinant as sum of two or more determinants)

Chapter 4 Class 12 Determinants

Concept wise

- Finding determinant of a 2x2 matrix
- Evalute determinant of a 3x3 matrix
- Area of triangle
- Equation of line using determinant
- Finding Minors and cofactors
- Evaluating determinant using minor and co-factor
- Find adjoint of a matrix
- Finding Inverse of a matrix
- Inverse of two matrices and verifying properties
- Finding inverse when Equation of matrice given
- Checking consistency of equations
- Find solution of equations- Equations given
- Find solution of equations- Statement given
- Verifying properties of a determinant
- Two rows or columns same
- Whole row/column zero
- Whole row/column one
- Making whole row/column one and simplifying
- Proving Determinant 1 = Determinant 2
- Solving by simplifying det.
- Using Property 5 (Determinant as sum of two or more determinants)

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.