WMC 2018 Senior Problem 12

Algebra Level 2

If the determinant [ a b a b a b a b 1 ] = 0 \left[\begin{matrix}a&b&a\\b&a&b\\a&b&1\\\end{matrix}\right]=0 , then what is the complete solution?

None of these a = b a=b or a = − 1 a=-1 a = 1 a=1 or a = b a=b a = 1 a=1 or a = − b a=-b a = − 1 a=-1 or a = − b a=-b

Tiger Ang by Tiger Ang , 20, Malaysia

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1 solution

X X
Oct 14, 2018

a 2 − a b 2 − b 2 + a b 2 + a b 2 − a 3 = 0 = − ( a − 1 ) ( a − b ) ( a + b ) a^2-ab^2-b^2+ab^2+ab^2-a^3=0=-(a-1)(a-b)(a+b)

So, the complete solution is a = 1 a=1 or a = b a=b or a = − b a=-b

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