If a + b = -2 then show that `|[1,1,1],[a,b,c],[a^3, b^3, c^3]|` = 0

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The determinant of a matrix `[[a, b, c],[d,e,f],[g,h,i]]` is given by:

`|[a, b, c],[d,e,f],[g,h,i]|= a*(e*i - f*g) - b*(d*i - g*f) + c*(d*h -e*g)`

`|[1,1,1],[a,b,c],[a^3, b^3, c^3]|`

= `(b*c^3 - b^3*c) - (a*c^3 - c*a^3) + (a*b^3 - a^3*b) `

=> `b*c^3 - b^3*c - a*c^3 + c*a^3 + a*b^3 - a^3*b `

=> `(a - b)(b - c)(c - a)(a + b + c) `

This is not equal to 0 if a + b = -2.

The determinant of the given matrix is not equal to 0 if a + b = -2.

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If a + b was instead equal to -c, the determinant would be 0.

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