Take α,β,γfromR1,R2andR3
Δ=αβγ∣∣
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∣∣1+a1b1αa1b2αa1b3α0+a2b1β1+a2b2γ1+a3b3γ0+a3b1βa3b2γ1+a3b3γ∣∣
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Split into two determinants say Δ1+Δ2
Δ1=1∣∣
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∣∣1+a2b2βa2b3βa2b3γ1+a3b3γ∣∣
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=1+a2b2β+a3b3γ
Δ2=b1∣∣
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∣∣a1αa1b2αa1b3αa2β1+a2b2βa2b3βa3γa3b2γ1+a3b3γ∣∣
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Apply C2−b2C1,C3−b3C1
Δ2=b1∣∣
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∣∣a1α00a2β10a3γ01∣∣
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∣∣=a1b1α
∴Δ=αβγ[1+a1b1α+a2b2β+a3b3γ]