Misc 1

Transcript

Misc 1 Prove that the determinant x﷮ sin﷮θ﷯﷮ cos﷮θ﷯﷮− sin﷮θ﷯﷮−x﷮1﷮ cos﷮θ﷯﷮1﷮x﷯﷯ is independent of θ. Let ∆ = x﷮ sin﷮θ﷯﷮ cos﷮θ﷯﷮− sin﷮θ﷯﷮−x﷮1﷮ cos﷮θ﷯﷮1﷮x﷯﷯ ∆ = x −𝑥﷮1﷮1﷮𝑥﷯﷯ – sinθ − sin﷮θ﷯﷮1﷮ cos﷮θ﷯﷮𝑥﷯﷯ + cos θ − sin﷮θ﷯﷮−𝑥﷮ cos﷮θ﷯﷮1﷯﷯ = x ( – x2 – 1) – sinθ ( – xsinθ – cos θ) + cos θ ( – sinθ + x cos θ) = – x3 – x + x. sin﷮2θ﷯ + sin﷮θ﷯ cos θ – sin θ cos θ + x cos2θ = – x3 – x + x sin2 θ + x+ cos2 θ + sin θ cos θ – sin θ cos θ = – x3 – x + x (sin2 θ + cos2 θ ) + 0 = – x3 – x + x (1) = – x3 Hence ∆ = – x3 Which has no θ term Thus, the determinant is independent of θ Hence Proved