# Consider the matrix A = [[1,a,a^2],[1,b,b^2],[1,c,c^2]] Show that A is invertible if a, b, c are different numbers.

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Expert Answers

mathsworkmusic | Certified Educator

For A to be invertible its determinant must be zero, since

`A^(-1) = 1/det(A)C^T`

where `C` is the cofactor matrix of A. If det(A) is zero, then we are dividing by zero and the inverse matrix is thus undefined.

Through calculation we find that det(A) = |A| equals

(b-a)(c-a)(c-b).

If any element of this is zero, then the determinant of A is zero and A is non-invertible. Therefore we must have `a != b`, `a !=c` and `b!=c` .

In other words, **for the matrix A to be invertible none of a, b and c can be the same numbers.**