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Last updated at Jan. 23, 2020 by Teachoo

Transcript

Misc 2 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.

Proving Determinant 1 = Determinant 2

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.