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Question

Show that ∣ ∣b+cabac+abcba+bcac∣ ∣=3abca3b3c3.

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Solution

The given determinant
∣ ∣b+cabac+abcba+bcac∣ ∣
=∣ ∣baacbbacc∣ ∣∣ ∣bbaccbaac∣ ∣+∣ ∣caaabbbcc∣ ∣∣ ∣cbaacbbac∣ ∣.
Of these four determinants the first three vanish, as two columns are same. Thus the expression reduces to the last of the four determinants; hence its value
={c(c2ab)b(acb2)+a(a2bc)}
=3abca3b3c3.

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