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Question 8 - Whole row/column one - Chapter 4 Class 12 Determinants
Last updated at Dec. 16, 2024 by Teachoo
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)
Transcript
Question 8 Evaluate ∆ = 1𝑎𝑏𝑐1𝑏𝑐𝑎1𝑐𝑎𝑏 ∆ = 1𝑎𝑏𝑐1𝑏𝑐𝑎1𝑐𝑎𝑏 We need to obtain 0 in 2nd row as well as 3rd row Applying R2 → R2 – R1 ∆ = 1𝑎𝑏𝑐𝟏 –𝟏𝑏−𝑎𝑐𝑎−𝑏𝑐1𝑐𝑎𝑏 = 1𝑎𝑏𝑐𝟎𝑏−𝑎𝑐(𝑎−𝑏)1𝑐𝑎𝑏 Applying R3 → R3 – R1 = 1𝑎𝑏𝑐0𝑏−𝑎𝑐(𝑎−𝑏)𝟏−𝟏𝑐−𝑎𝑎𝑏−𝑏𝑐 = 1𝑎𝑏𝑐0(𝑏−𝑎)𝑐(𝑎−𝑏)𝟎(𝑐−𝑎)𝑏(𝑎−𝑐) Expanding it along C1 = 1 𝑏−𝑎𝑐 𝑎−𝑏 𝑐−𝑎𝑏 𝑎−𝑐–0 𝑎𝑏𝑐 𝑐−𝑎𝑏 𝑎−𝑐 +0 𝑎𝑏𝑐 𝑏−𝑎𝑐 𝑎−𝑏 = 1 b−𝑎c 𝑎−𝑏 c−a𝑏 𝑎−𝑐 – 0 + 0 = 1 (b – a) b(a – c) – (c – a) (c) (a – b) = –(a – b) b ( – (c – a)) – (c – a) c (a – b) = (a – b) b (c – a) – (c – a) c (a – b) = (a – b)(c – a) (b – c)) = (a – b) (b – c) (c – a) Thus ∆ = (a – b) (b – c) (c – a)
