# Ex 4.4, 4 - Chapter 4 Class 12 Determinants

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Ex4.4, 4 Using Cofactors of elements of third column, evaluate ∆ = 1xyz1yzx1zxy ∆ = 1xyz1yzx1zxy ∆ = a13 A13 + a23 A23 + a33 A33 a13 = yz , a23 = zx , a33 = xy , Calculating cofactors of third column i.e. A13 , A23 , And A33 M13 = 1xyz1yzx1zxy = 1y1𝑧 = 1 × z – 1 × y = z – y M23 = 1xyz1yzx1zxy = 1x1z = 1 × z – 1 × x = z – x M33 = 1xyz1yzx1zxy = 1x1𝑦 = 1 × y – 1 × x = y – x A13 = ( – 1)1+3 M13 = ( – 1)4 . (z – y) = z – y A23 = ( – 1)2+3 . M23 = ( – 1)5 . (z – x) = ( – 1) (z – x) = x – z A33 = (– 1)3 + 3 . M33 = (– 1)6 . M33 = 1 . (y – x) = y – x Now, ∆ = a13 A13 + a23 A23 + a33 A33 = yz (z – y) + zx (x – z) + xy (y – x) = yz2 – y2z + zx2 – z2x + xy2 – x2y = (yz2 – y2z) + (xy2 – z2x) + (zx2 – x2y) = yz (z – y) + x (y2 – z2) + x2 (z – y) = – yz (y – z) + x (y2 – z2) – x2 (y – z) = – yz (y – z) + x (y + z) (y – z) – x2 (y – z) = (y – z) −𝑦𝑧+𝑥 𝑦+𝑧−𝑥2 = (y – z) −𝑦𝑧+𝑥𝑦+𝑥𝑧−𝑥2 = (y – z) 𝑧 𝑥−𝑦+𝑥(𝑦−𝑥) = (y – z) 𝑧 𝑥−𝑦−𝑥(𝑥−𝑦) = (y – z) 𝑧−𝑥 (𝑥−𝑦) = (x – y) (y – z) (z – x)

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.