# Example 29 - Chapter 4 Class 12 Determinants

Last updated at Sept. 21, 2018 by Teachoo

Last updated at Sept. 21, 2018 by Teachoo

Transcript

Example 29 The sum of three numbers is 6. If we multiply third number by 3 and add second number to it, we get 11. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method. Let the first, second & third number be x, y, z respectively Given, x + y + z = 6 y + 3z = 11 x + z = 2y or x 2y + z = 0 Step 1 Write equation as AX = B 1 1 1 0 1 3 1 2 1 = 6 11 0 Hence A = 1 1 1 0 1 3 1 2 1 , X = & B = 6 11 0 Step 2 Calculate |A| = 1 1 1 0 1 3 1 2 1 = 1 (1 + 6) 0 (1 + 2) + 1 (3 1) = 7 + 2 = 9 So, |A| 0 The system of equation is consistent & has a unique solutions Now, AX = B X = A-1 B Hence A = 1 1 1 0 1 3 1 2 1 , X = & B = 6 11 0 = 1 (1 + 6) 0 (1 + 2) + 1 (3 1) = 7 + 2 = 9 0 Since determinant is not equal to O, A 1 exists Now find adj (A) adj (A) = 11 12 13 21 22 23 31 32 33 = 11 21 31 12 22 32 13 32 33 Now, AX = B X = A-1 B Step 3 Calculating X = A-1 B Calculating A-1 Now, A-1 = 1 |A| adj (A) adj A = A11 A12 A13 A21 A22 A23 A31 A32 A33 = A11 A21 A31 A12 A22 A32 A13 A23 A33 A = 1 1 2 3 4 5 2 1 3 11 = 1 1 3 ( 2) = 1 + 6 = 7 12 = 0 1 3 1 = ( 3) = 3 13 = 0 2 1 1= 1= 21 = 1 1 2 1 = 1+2 = 3 22 = 1 1 1 1 = 1 1 = 0 23 = 1 2 1 1 = 2 1 = 3 = 3 31 = 1 3 1 1 = 3 1 = 2 32 = 1 3 0 1 = 3 0 = 3 33 = 1 1 1 0 = 1 0 = 1 Hence, adj (A) = 7 3 2 3 0 3 1 3 1 Now, A 1 = 1 adj (A) A 1 = 1 9 7 3 2 3 0 3 1 3 1 Solution of given system of equations is X = A 1 B = 1 9 7 3 2 3 0 3 1 3 1 6 11 0 = 1 9 42 33+0 18+0+0 6+33+0 = 1 9 9 18 27 = 1 2 3 x = 1, y = 2, z = 3

Find solution of equations- Statement given

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.