Without expanding at any stage show that x 2 -x x-1 x-2 2 x 2 -3x-1 3x 3x-3 x 2 -2x-3 2x-1 2x-1 -xA-Bwhere A and B are determinants of order 3 not involving x-

Apply R_ 2 (R_ 1 +R_ 3 ) Δ =| x ^ 2 +x amp; x+1 amp; x 2 4 amp; 0 amp; 0 x ^ 2 +2x+3 amp; 2x 1 amp; 2x 1 | Now apply ...

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Properties of Determinants

Without expan...

Question

Without expanding at any stage show that
$∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + x & x + 1 & x - 2 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = x A + B$
where $A$ and $B$ are determinants of order $3$ not involving $x$ .

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∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + x & x + 1 & x - 2 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

A x^{n} + B

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∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + x & x + 1 & x + 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = A x + B

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∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + x & x + 1 & x - 2 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

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∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + x & x + 1 & x - 2 {2 x}^{2} + 3 x - 1 & 3 x & 3 x - 3 x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = a x - 12,

a^{2}

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