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Question

If D=∣ ∣abccabbca∣ ∣ and D=∣ ∣b+cc+aa+ba+bb+cc+ac+aa+bb+c∣ ∣ then D=

A
2D
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B
D
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C
3D
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D
6D
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Solution

The correct option is B 2D
D=∣ ∣b+cc+aa+ba+bb+cc+ac+aa+bb+c∣ ∣
D=∣ ∣bc+aa+bab+cc+aca+bb+c∣ ∣+∣ ∣cc+aa+bbb+cc+aaa+bb+c∣ ∣
D=∣ ∣bca+babc+acab+c∣ ∣+∣ ∣baa+bacc+acbb+c∣ ∣+∣ ∣cca+bbbc+aaab+c∣ ∣+∣ ∣caa+bbcc+aabb+c∣ ∣
D=∣ ∣bca+babc+acab+c∣ ∣+∣ ∣baa+bacc+acbb+c∣ ∣+∣ ∣caa+bbcc+aabb+c∣ ∣ (Since,a determinant with two identical rows/columns has a value 0)
D=∣ ∣bcaabccab∣ ∣+∣ ∣bcbabacac∣ ∣+∣ ∣baaacccbb∣ ∣+∣ ∣babacacbc∣ ∣+∣ ∣caabccabb∣ ∣+∣ ∣cabbcaabc∣ ∣
D=∣ ∣bcaabccab∣ ∣+∣ ∣cabbcaabc∣ ∣
=∣ ∣abccabbca∣ ∣+∣ ∣abccabbca∣ ∣ (On interchanging rows , determinant value changes by -sign)
D=2D

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