First slide

Introduction to Determinants

Question

If $|\begin{array}{ccc} yz - x^{2} & zx - y^{2} & xy - z^{2} \\ xz - y^{2} & xy - z^{2} & yz - x^{2} \\ xy - z^{2} & yz - x^{2} & zx - y^{2} \end{array}| = |\begin{array}{ccc} r^{2} & u^{2} & u^{2} \\ u^{2} & r^{2} & u^{2} \\ u^{2} & u^{2} & r^{2} \end{array}|,$ then

Moderate

A

$r^{2} = x + y + z$
B

$r^{2} = x^{2} + y^{2} + z^{2}$
C

$u^{2} = yz + zx + xy$
D

$u^{2} = xyz$

Solution

In the left-hand determinant, each element is the cofactor of the elements of the determinant
$|\begin{array}{lll} x y z \\ y z x \\ z x y \end{array}| = Δ^{} (say)$
Hence,
$\begin{array}{l} Δ^{^{2}} = |\begin{array}{lll} x & y & z \\ y & z & x \\ z & x & y \end{array}| |\begin{array}{ccc} x & y & z \\ y & z & x \\ z & x & y \end{array} ∣| \\ = |\begin{array}{ccc} x^{2} + y^{2} + z^{2} & xy + yz + zx & xz + yx + zy \\ Σxy & {Σx}^{2} & Σxy \\ Σxy & Σxy & {Σx}^{2} \end{array}| \\ = |\begin{array}{ccc} r^{2} & u^{2} & u^{2} \\ u^{2} & r^{2} & u^{2} \\ u^{2} & u^{2} & r^{2} \end{array}| [∵ x^{2} + y^{2} + z^{2} = r^{2}, xy + yz + zx = u^{2}] \end{array}$

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