Consider the matrix A = [[1,a,a^2],[1,b,b^2],[1,c,c^2]] Show that det(A) = (b -a)(c -a)(c -b) using row reduction.

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mathsworkmusic | (Level 2) Educator

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We reduce A by making linear row operations

 

`A = [[1,a,a^2],[1,b,b^2],[1,c,c^2]]`

`= [[1,a,a^2],[0,b-a, b^2-a^2],[0,c-a,c^2-a^2]]`     (subtracting row 1 from rows 2 and 3).

``Now, note that

`det(A) = |A| = A_(11)|[b-a,b^2-a^2],[c-a,c^2-a^2]|`

` ` `- A_(22)|[0,b^2-a^2],[0,c^2-a^2]| + A_(33)|[0,b-a],[0,c-a]|`

`= (b-a)(c^2-a^2)- (b^2-a^2)(c-a)`

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

`= (b-a)(c-a)[(c+a)-(b+a)] = (b-a)(c-a)(c-b)`` `

```required`

 

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