Misc 9

Transcript

Misc 9 Evaluate 𝑥﷮𝑦﷮𝑥+𝑦﷮𝑦﷮𝑥+𝑦﷮𝑥﷮𝑥+𝑦﷮𝑥﷮𝑦﷯﷯ Let ∆ = 𝑥﷮𝑦﷮𝑥+𝑦﷮𝑦﷮𝑥+𝑦﷮𝑥﷮𝑥+𝑦﷮𝑥﷮𝑦﷯﷯ Applying R1→ R1 + R2 + R3 = 𝑥+𝑦+𝑥+𝑦﷮𝑦+𝑥+𝑦+𝑥﷮𝑥+𝑦+𝑥+𝑦﷮𝑦﷮𝑥+𝑦﷮𝑥﷮𝑥+𝑦﷮𝑥﷮𝑦﷯﷯ = 2x+2y﷮2x+2y﷮2x+2y﷮y﷮x+y﷮x﷮x+y﷮x﷮y﷯﷯ = 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) 1﷮1﷮1﷮y﷮x+y﷮x﷮x+y﷮x﷮y﷯﷯ Applying C2→ C2 – C1 = 2(x+y) 1﷮𝟏−𝟏﷮1﷮y﷮x+y−𝑦﷮x﷮x+y﷮x−x−y﷮y﷯﷯ = 2(x+y) 1﷮𝟎﷮1﷮y﷮x﷮x﷮x+y﷮−y﷮x﷯﷯ Applying C3 →C3 – C1 = 2(x+y) 1﷮0﷮𝟏−𝟏﷮y﷮x﷮x−y﷮x+y﷮−y﷮y−x−y﷯﷯ = 2(x+y) 1﷮0﷮𝟎﷮y﷮x﷮x−y﷮x+y﷮−y﷮−x﷯﷯ Expanding determinant along R1 = 2(x+y) 1 𝑥﷮𝑥−𝑦﷮−𝑦﷮−𝑥﷯﷯−0 𝑦﷮𝑥−𝑦﷮𝑥+𝑦﷮−𝑥﷯﷯+0 𝑦﷮𝑥﷮𝑥+𝑦﷮−𝑦﷯﷯﷯ = 2(x+y) 1 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)