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Engineering

Mathematics

Introduction to Determinants

Question

$|\begin{array}{ccc} 0 & xyz & x - z \\ y - x & 0 & y - z \\ z - x & z - y & 0 \end{array}| =$ _________

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Let $D = ∣ \begin{array}{ccc} 0 & xyz & x - z \\ y - x & 0 & y - z \\ z - x & z - y & 0 \end{array} ∣$

R₂ – R₃ → R₂

∴ D = |\begin{array}{ccc} 0 & xyz & x - z \\ y - z & y - z & y - z \\ z - x & z - y & 0 \end{array}|

∴ D = (y - z) |\begin{array}{ccc} 0 & xyz & x - z \\ 1 & 1 & 1 \\ z - x & z - y & 0 \end{array}|

∴ D = (y – z){– xyz [0 – (z – x)] + (x – z)[(z – y) – (z – x)}

∴ D = (y – z){– xyz[x – z] + (x – z)[z – y – z + x]}

∴ D = (y – z){– xyz[x – z] + (x – z)[x – y]}

∴ D = (y – z)(x – z){ – xyz + x – y}

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