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Question

Without expanding the determinants, prove that
∣ ∣abcxyzpqr∣ ∣=∣ ∣ybqxapzcr∣ ∣=∣ ∣xyzpqrabc∣ ∣.

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Solution

∣ ∣abcxyzpqr∣ ∣=∣ ∣ybqxapzcr∣ ∣∣ ∣xyzpqrabc∣ ∣Δ1=Δ2=Δ3Δ2=∣ ∣ybqxapzcr∣ ∣
Taking transpose as |A|=AT
Δ2=∣ ∣yxzbacqzr∣ ∣
Interchanging C1 and C2
Δ2=∣ ∣xyzabcpqr∣ ∣
Interchanging R1 and R2
Δ2=∣ ∣abcxyzpqr∣ ∣......(i)
Δ3=∣ ∣xyzpqrabc∣ ∣
Interchanging R1 and R3
Δ3=∣ ∣abcpqrxyz∣ ∣
Interchanging R2 and R3
Δ3=∣ ∣abcxyzpqr∣ ∣.....(ii)
From (i) and (ii)
Δ1=Δ2=Δ3
Hence proved.

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