Check sibling questions

Example 15  - If x, y, z are different, show 1 + xyz = 0 - Class 12

Example 15 - Chapter 4 Class 12 Determinants - Part 2
Example 15 - Chapter 4 Class 12 Determinants - Part 3
Example 15 - Chapter 4 Class 12 Determinants - Part 4
Example 15 - Chapter 4 Class 12 Determinants - Part 5

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Example 15 If x, y, z are different and Δ = |■8(x&x2&1+x3@y&y2&1+y3@z&z2&1+z3)| = 0 , then show that 1 + xyz = 0 Solving ∆ = |■8(x&x2&1+x3@y&y2&1+y3@z&z2&1+z3)| Here, expanding elements of C3 into two determinants = |■8(x&x2&1@y&y2&1@z&z2&1)| + |■8(x&x2&x3@y&y2&y3@z&z2&z3)| = |■8(x&x2&1@y&y2&1@z&z2&1)| + |■8(x&x2&x3@y&y2&y3@z&z2&z3)| = |■8(x&x2&1@y&y2&1@z&z2&1)|+ xyz |■8(1&x&x2@1&y&y2@1&z&z2)| = (−1) |■8(x&1&x2@y&1&y2@z&1&z2)| + xyz |■8(1&x&x2@1&y&y2@1&z&z2)| = (−1)(−1)|■8(1&x&x2@1&y&y2@1&z&z2)| + xyz |■8(1&x&x2@1&y&y2@1&z&z2)| Taking x , y , z common from R1, R2, R3 respectively Replacing C3↔ C2 Replacing C1↔ C2 If any two columns of a determinant are interchanged , then sign of determinant changes = |■8(1&x&x2@1&y&y2@1&z&z2)| + xyz |■8(1&x&x2@1&y&y2@1&z&z2)| = |■8(1&x&x2@1&y&y2@1&z&z2)| (1 + xyz) Using R2 → R2 – R1 and R3 → R3 – R1 = |■8(1&x&x2@𝟏 −𝟏&y−x&y2 −x2@𝟏−𝟏&z−x&z2 −x2)| (1+ xyz) = |■8(1&x&x2@𝟎&(y−x)&(y −x)(y+x)@𝟎&(z−x)&(z −x)(z+x))| (1+ xyz) Taking common factor (y – x) from R2 & (z – x) from R3 = (1 + xyz) (y – x) (z – x) |■8(1&x&x2@0&1&y+x@0&1&z+x)| Expanding determinant = (1 + xyz) (y – x) (z – x) (z – y) (1|■8(1&𝑦+𝑥@1&𝑧+𝑥)|" – 0 " |■8(𝑥&𝑥2@1&𝑧+𝑥)|" + 0" |■8(𝑥&𝑥2@1&𝑦+𝑥)|) = (1 + xyz) (y – x) (z – x) (1 (y + x) – (y + x) + 0 + 0) = (1 + xyz) (y – x) (z – x) (z + y – y – x) = (1 + xyz) (y – x) (z – x) (z – y) ∴ ∆ = (1 + xyz) (y – x) (z – x) (z – y) Since ∆ = 0 given (1 + xyz) (y – x) (z – x) (z – y) = 0 Since it is given that x, y, z all are different, i.e., y – x ≠ 0, z – x ≠ 0, z – y ≠ 0, So, only Possibility is (1 + xyz) = 0 Hence Proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.