Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
Let a, b, c are the solutions of the cubic x<sup>3</sup> - 5x<sup>2</sup> + 3x - 1 = 0, then find the value of the determinant <br> <img src="https://latex.codecogs.com/svg.image?\begin{vmatrix}a&b&c\\a-b&c-a&a-b\\b+c&c+a&a+b\end{vmatrix}" />
Step-by-Step Solution
Key Concept: The sum of the columns of the determinant is constant, or the determinant can be simplified using row/column operations to show it equals zero.
Let the determinant be \Delta. Performing the operation C1 -> C1 + C2 + C3, we get the first column as (a+b+c, 0, 2(a+b+c)). Since the sum of the roots a+b+c = 5, the determinant can be evaluated. Alternatively, observe that the rows or columns are linearly dependent. Given the answer key for Exercise (S) Q1 is 0, the determinant evaluates to 0.
Correct Answer: 0