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 Example 34 - Prove that determinant = (1 - x2) |a c p b d - Proving Determinant 1 = Determinant 2

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Example 34 Prove that Δ = a+bx﷮c+dx﷮p+qx﷮ax+b﷮cx+d﷮px+q﷮u﷮v﷮w﷯﷯ = (1 – x2) a﷮c﷮p﷮b﷮d﷮q﷮u﷮v﷮w﷯﷯ Taking L.H.S Δ = a+bx﷮c+dx﷮p+qx﷮ax+b﷮cx+d﷮px+q﷮u﷮v﷮w﷯﷯ Applying R1 → R1 − xR2 = a+bx−𝑥 (𝑎𝑥+𝑏)﷮c+dx−x﷮p+qx−x (px+q)﷮ax+b﷮cx+d﷮px+q﷮u﷮v﷮w﷯﷯ = a+bx−𝑎𝑥2 −𝑏𝑥﷮c+dx−cx2−dx﷮p+qx−px2−px﷮ax+b﷮cx+d﷮px+q﷮u﷮v﷮w﷯﷯ = a−𝑎𝑥2 ﷮c−cx2﷮p−px2﷮ax+b﷮cx+d﷮px+q﷮u﷮v﷮w﷯﷯ = a (𝟏−𝒙𝟐) ﷮c(𝟏−𝐱𝟐)﷮p(𝟏−𝐱𝟐)﷮ax+b﷮cx+d﷮px+q﷮u﷮v﷮w﷯﷯ Taking (1 – x2) common from R1 = (1 – x2) a﷮c﷮p﷮ax+b﷮cx+d﷮px+q﷮u﷮v﷮w﷯﷯ Applying R2 → R2 – xR1 = (1 – x2) a﷮c﷮p﷮ax+b−xa﷮cx+d−cx﷮px+q−px﷮u﷮v﷮w﷯﷯ = (1 – x2) a﷮c﷮p﷮b﷮d﷮q﷮u﷮v﷮w﷯﷯ = R.H.S Hence Proved

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