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Question

Evaluate: ∣ ∣ ∣23+35546+15510115+3155∣ ∣ ∣

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Solution

=∣ ∣ ∣235546510115155∣ ∣ ∣+∣ ∣ ∣355155103155∣ ∣ ∣

=∣ ∣ ∣23552×23552×55×235×355∣ ∣ ∣+∣ ∣ ∣355355525333555∣ ∣ ∣

=∣ ∣ ∣23552×23552×55×235×355∣ ∣ ∣+∣ ∣ ∣3553×55525333555∣ ∣ ∣

=∣ ∣ ∣235522355255235355∣ ∣ ∣+∣ ∣ ∣355355525333555∣ ∣ ∣
In the first determinant using C1C123,C2C25 and C3C25
In the second determinant using C1C13,C2C25 and C3C25

=2355∣ ∣ ∣111252535∣ ∣ ∣+355∣ ∣ ∣111552335∣ ∣ ∣

C1C1C3 in the first determinant and

C1C1C2

523∣ ∣ ∣011052035∣ ∣ ∣+53∣ ∣ ∣011052035∣ ∣ ∣

=0 since one column in both the determinants are zero
Hence Proved


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