Example 32 - Show that Determinant = 2xyz (x +y+z)3 - Solving by simplifying det.

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  1. Class 12
  2. Important Question for exams Class 12

Transcript

Example 32 Show that ฮ” = |โ– 8((๐‘ฆ+๐‘ง)2&๐‘ฅ๐‘ฆ&๐‘ง๐‘ฅ@๐‘ฅ๐‘ฆ&(๐‘ฅ+๐‘ง)2&๐‘ฆ๐‘ง@๐‘ฅ๐‘ง&๐‘ฆ๐‘ง&(๐‘ฅ+๐‘ฆ)2)| = 2xyz (x + y + z)3 Taking L.H.S ฮ” = |โ– 8((๐‘ฆ+๐‘ง)3&๐‘ฅ๐‘ฆ&๐‘ง๐‘ฅ@๐‘ฅ๐‘ฆ&(๐‘ฅ+๐‘ง)2&๐‘ฆ๐‘ง@๐‘ฅ๐‘ง&๐‘ฆ๐‘ง&(๐‘ฅ+๐‘ฆ)2)| Divide & Multiply by xyz = ๐‘ฅ๐‘ฆ๐‘ง/๐‘ฅ๐‘ฆ๐‘ง |โ– 8((๐‘ฆ+๐‘ง)3&๐‘ฅ๐‘ฆ&๐‘ง๐‘ฅ@๐‘ฅ๐‘ฆ&(๐‘ฅ+๐‘ง)2&๐‘ฆ๐‘ง@๐‘ฅ๐‘ง&๐‘ฆ๐‘ง&(๐‘ฅ+๐‘ฆ)2)| = 1/๐‘ฅ๐‘ฆ๐‘ง x. y. z |โ– 8((๐‘ฆ+๐‘ง)3&๐‘ฅ๐‘ฆ&๐‘ง๐‘ฅ@๐‘ฅ๐‘ฆ&(๐‘ฅ+๐‘ง)2&๐‘ฆ๐‘ง@๐‘ฅ๐‘ง&๐‘ฆ๐‘ง&(๐‘ฅ+๐‘ฆ)2)| = 1/๐‘ฅ๐‘ฆ๐‘ง x. y. z |โ– 8((๐‘ฆ+๐‘ง)3&๐‘ฅ๐‘ฆ&๐‘ง๐‘ฅ@๐‘ฅ๐‘ฆ&(๐‘ฅ+๐‘ง)2&๐‘ฆ๐‘ง@๐‘ฅ๐‘ง&๐‘ฆ๐‘ง&(๐‘ฅ+๐‘ฆ)2)| Multiplying R1 by x , R2 by y & R3 by z = 1/๐‘ฅ๐‘ฆ๐‘ง |โ– 8(๐’™(๐‘ฆ+๐‘ง)2&๐’™(๐‘ฅ๐‘ฆ)&๐’™(๐‘ง๐‘ฅ)@๐’š(๐‘ฅ๐‘ฆ)&๐’š(๐‘ฅ+๐‘ง)2&๐’š(๐‘ฆ๐‘ง)@๐’›(๐‘ฅ๐‘ง)&๐‘ฆ๐’›2&๐’›(๐‘ฅ+๐‘ฆ)2)| Taking out x common from C1, y common from C2 & z common from C3 = ๐‘ฅ๐‘ฆ๐‘ง/๐‘ฅ๐‘ฆ๐‘ง |โ– 8((y+z)2&x2&x2@y2&(x+z)2&y2@๐‘ง2&z2&(x+y)2)| Applying C2โ†’ C2 โ€“ C1 = |โ– 8((y+z)2&x2โˆ’(๐‘ฆ+๐‘ง)2&x2@y2&(x+z)2โˆ’y2&y2@๐‘ง2&z2โˆ’z2&(x+y)2)| = |โ– 8((y+z)2&(xโˆ’(๐‘ฆ+๐‘ง))(๐‘ฅ+(๐‘ฆ+๐‘ง))&x2@y2&((x+z)โˆ’๐‘ฆ)(๐‘ฅ+๐‘ง+๐‘ฆ)&y2@๐‘ง2&0&(x+y)2)| = |โ– 8((y+z)2&(๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง)(๐’™+๐’š+๐’›)&x2@y2&(x+zโˆ’y)(๐’™+๐’š+๐’›)&y2@๐‘ง2&0&(x+y)2)| Taking out (๐‘ฅ+๐‘ฆ+๐‘ง) common from C2 = (๐‘ฅ+๐‘ฆ+๐‘ง)|โ– 8((y+z)2&๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง&x2@y2&๐‘ฅ+๐‘งโˆ’๐‘ฆ&y2@๐‘ง2&0&(x+y)2)| Applying C3 โ†’ C3 โ€“ C1 = (๐‘ฅ+๐‘ฆ+๐‘ง)|โ– 8((y+z)2&๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง&๐‘ฅ2 โˆ’(๐‘ฆ+๐‘ง)2@y2&๐‘ฅ+๐‘งโˆ’๐‘ฆ&๐‘ฆ2โˆ’๐‘ฆ2@๐‘ง2&0&(๐‘ฅ+๐‘ฆ)2โˆ’๐‘ง2)| = (๐‘ฅ+๐‘ฆ+๐‘ง)|โ– 8((y+z)2&๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง&(๐’™+๐’š+๐’›)(๐‘ฅโˆ’(๐‘ฆ+๐‘ง))@y2&๐‘ฅ+๐‘งโˆ’๐‘ฆ&0@๐‘ง2&0&(๐’™+๐’š+๐’›)((๐‘ฅ+๐‘ฆ)โˆ’๐‘ง))| Taking out (๐‘ฅ+๐‘ฆ+๐‘ง) Common from C3 = (๐‘ฅ+๐‘ฆ+๐‘ง)(๐‘ฅ+๐‘ฆ+๐‘ง)|โ– 8((y+z)2&๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง&๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง@y2&๐‘ฅ+๐‘งโˆ’๐‘ฆ&0@๐‘ง2&0&๐‘ฅ+๐‘ฆโˆ’๐‘ง)| = (๐‘ฅ+๐‘ฆ+๐‘ง)2 |โ– 8((y+z)2&๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง&๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง@y2&๐‘ฅ+๐‘งโˆ’๐‘ฆ&0@๐‘ง2&0&๐‘ฅ+๐‘ฆโˆ’๐‘ง)| Applying R1โ†’ R1 โ€“ R2 โ€“ R3 = (๐‘ฅ+๐‘ฆ+๐‘ง)2|โ– 8((y+z)2โˆ’๐‘ฆ2โˆ’๐‘ง2&(๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง)โˆ’(๐‘ฅ+๐‘งโˆ’๐‘ฆ)โˆ’0&(๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง)โˆ’0โˆ’(๐‘ฅ+๐‘ฆโˆ’๐‘ง)@y2&๐‘ฅ+๐‘งโˆ’๐‘ฆ&0@๐‘ง2&0&๐‘ฅ+๐‘ฆโˆ’๐‘ง)| = (๐‘ฅ+๐‘ฆ+๐‘ง)2|โ– 8(๐‘ฆ2+๐‘ง2+2๐‘ฆ๐‘งโˆ’๐‘ฆ2โˆ’๐‘ง2&๐‘ฅโˆ’๐‘ฅโˆ’๐‘ฆ+๐‘ฆโˆ’๐‘งโˆ’๐‘ง&๐‘ฅโˆ’๐‘ฅโˆ’๐‘ฆโˆ’๐‘ฆโˆ’๐‘ง+๐‘ง@y2&๐‘ฅ+๐‘งโˆ’๐‘ฆ&0@๐‘ง2&0&๐‘ฅ+๐‘ฆโˆ’๐‘ง)| = (๐‘ฅ+๐‘ฆ+๐‘ง)2|โ– 8(2๐‘ฆ๐‘ง&โˆ’2๐‘ง&โˆ’2๐‘ฆ@y2&๐‘ฅ+๐‘งโˆ’๐‘ฆ&0@๐‘ง2&0&๐‘ฅ+๐‘ฆโˆ’๐‘ง)| Applying C2โ†’ C2 + 1/๐‘ฆ C1 = (๐‘ฅ+๐‘ง+๐‘ฆ)2|โ– 8(2๐‘ฆ๐‘ง&โˆ’2๐‘ง+๐Ÿ/๐’š(๐Ÿ๐’š๐’›)&2๐‘ฆ@y2&xโˆ’๐‘ฆ+๐‘ง+๐Ÿ/๐’š (๐’š๐Ÿ)&0@๐‘ง2&0+๐Ÿ/๐’š(๐’›๐Ÿ)&๐‘ฅ+๐‘ฆโˆ’๐‘ง)| = (๐‘ฅ+๐‘ง+๐‘ฆ)2|โ– 8(2๐‘ฆ๐‘ง&0&2๐‘ฆ@y2&x+๐‘ง&0@๐‘ง2&๐‘ง^2/๐‘ฆ&๐‘ฅ+๐‘ฆโˆ’๐‘ง)| Applying C3โ†’ C3 + 1/๐‘ง C1 = (๐‘ฅ+๐‘ฆ+๐‘ง)2|โ– 8(2๐‘ฆ๐‘ง&0&โˆ’2๐‘ฆ+๐Ÿ/๐’›(๐Ÿ๐’š๐’›)@y2&๐‘ฅ+๐‘ง&0+๐Ÿ/๐’› (๐’š๐Ÿ)@๐‘ง2&๐‘ง^2/๐‘ฆ&(๐‘ฅ+๐‘ฆโˆ’๐‘ง)+๐Ÿ/๐’› (๐’›๐Ÿ))| = (๐‘ฅ+๐‘ฆ+๐‘ง)2|โ– 8(2๐‘ฆ๐‘ง&0&0@y2&๐‘ฅ+๐‘ง&๐‘ฆ^2/๐‘ง @๐‘ง2&๐‘ง^2/๐‘ฆ&๐‘ฅ+๐‘ฆ)| Expanding Determinant along R1 = (๐‘ฅ+๐‘ฆ+๐‘ง)2(2๐‘ฆ๐‘ง|โ– 8(๐‘ฅ+๐‘ง&๐‘ง^2/๐‘ฆ@๐‘ง^2/๐‘ฆ&๐‘ฅ+๐‘ฆ)|โˆ’0|โ– 8(๐‘ฆ2&๐‘ฆ^2/๐‘ง@๐‘ง^2&๐‘ฅ+๐‘ฆ)|+0|โ– 8(๐‘ฆ2&๐‘ฅ+๐‘ฆ@๐‘ง^2&๐‘ง^2/๐‘ฆ)|) = (๐‘ฅ+๐‘ฆ+๐‘ง)2(2๐‘ฆ๐‘ง|โ– 8(๐‘ฅ+๐‘ง&๐‘ง^2/๐‘ฆ@๐‘ง^2/๐‘ฆ&๐‘ฅ+๐‘ฆ)|โˆ’0+0) = (๐‘ฅ+๐‘ฆ+๐‘ง)2 ("2yz " ("(x + z) (x + y) โ€“ " ๐‘ง2/๐‘ฆ " " (๐‘ฆ2/๐‘ง))" โ€“ 0 + 0" ) = (๐‘ฅ+๐‘ฆ+๐‘ง)2 (2yz ((x + z) (x + y) โ€“ zy ) = (๐‘ฅ+๐‘ฆ+๐‘ง)2 (2yz) ((x + z) (x + y) โ€“ zy ) = (๐‘ฅ+๐‘ฆ+๐‘ง)2 (2yz) (x2 + xy + zx + zy โ€“ zy) = (๐‘ฅ+๐‘ฆ+๐‘ง)2 (2yz) (x2 + xy + xz) = (๐‘ฅ+๐‘ฆ+๐‘ง)2 (2yz) . x (x + y + z) = (๐‘ฅ+๐‘ฆ+๐‘ง)3 (2xyz) = (2xyz) (๐‘ฅ+๐‘ฆ+๐‘ง)^3 = R.H.S Hence Proved

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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.