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Miscellaneous
Misc. 2 Important
Misc 3
Misc 4
Misc 5
Misc 6 Important
Misc 7 Important
Misc 8
Misc 9 You are here
Misc 10
Misc 11 Important
Misc 12 Important
Misc. 13
Misc 14
Misc. 15 Important
Misc. 16 Important
Misc 17 (MCQ) Important
Misc 18 (MCQ)
Misc 19 (MCQ) Important
Matrices and Determinants - Formula Sheet and Summary Important
Last updated at Jan. 23, 2020 by Teachoo
Misc 9 Evaluate |■8(𝑥&𝑦&𝑥+𝑦@𝑦&𝑥+𝑦&𝑥@𝑥+𝑦&𝑥&𝑦)| Let ∆ = |■8(𝑥&𝑦&𝑥+𝑦@𝑦&𝑥+𝑦&𝑥@𝑥+𝑦&𝑥&𝑦)| Applying R1→ R1 + R2 + R3 = |■8(𝑥+𝑦+𝑥+𝑦&𝑦+𝑥+𝑦+𝑥&𝑥+𝑦+𝑥+𝑦@𝑦&𝑥+𝑦&𝑥@𝑥+𝑦&𝑥&𝑦)| = |■8(2x+2y&2x+2y&2x+2y@y&x+y&x@x+y&x&y)| = |■8(𝟐(𝐱+𝐲)&𝟐(𝐱+𝐲)&𝟐(𝐱+𝐲)@y&x+y&x@x+y&x&y)| Taking common 2(x + y), from R1 = 𝟐(𝐱+𝐲) |■8(1&1&1@y&x+y&x@x+y&x&y)| Applying C2→ C2 – C1 = 2(x+y) |■8(1&𝟏−𝟏&1@y&x+y−𝑦&x@x+y&x−x−y&y)| = 2(x+y) |■8(1&𝟎&1@y&x&x@x+y&−y&y)| Applying C3 →C3 – C1 = 2(x+y) |■8(1&0&𝟏−𝟏@y&x&x−y@x+y&−y&y−(x+y))| = 2(x+y) |■8(1&0&𝟎@y&x&x−y@x+y&−y&−x)| Expanding determinant along R1 = 2(x+y) (1|■8(𝑥&𝑥−𝑦@−𝑦&−𝑥)|−0|■8(𝑦&𝑥−𝑦@𝑥+𝑦&−𝑥)|+0|■8(𝑦&𝑥@𝑥+𝑦&−𝑦)|) = 2(x+y) (1|■8(𝑥&𝑥−𝑦@−𝑦&−𝑥)|−0+0) = 2(x+y) (1( – x2 – ( –y) (x – y)) ) = 2(x+y) ( – x2 + y (x – y)) = 2(x+y) ( – x2 + xy – y2) = – 2(x+y) ( x2 + y2 – xy) = − 2(x3+y3) Hence , ∆ = – 2(𝐱𝟑+𝐲𝟑) (Using a3 + b3 = (a + b) (a2 + b2 – ab))