Let a, b, c are the solutions of the cubic x^3 - 5x^2 + 3x - 1 = 0, then find the value of the determinant <br> <math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced open='|' close='|'><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>a</mi><mo>-</mo><mi>b</mi></mtd><mtd><mi>c</mi><mo>-</mo><mi>a</mi></mtd><mtd><mi>c</mi><mo>-</mo><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi><mo>+</mo><mi>c</mi></mtd><mtd><mi>c</mi><mo>+</mo><mi>a</mi></mtd><mtd><mi>a</mi><mo>+</mo><mi>b</mi></mtd></mtr></mtable></mfenced></math>.
Step-by-Step Solution
Key Concept: Use the properties of determinants to simplify the expression. Specifically, perform row operations like R3 -> R3 + R1 and then use the fact that a, b, c are roots of the given cubic equation to evaluate the resulting determinant.
Let the determinant be \Delta. Performing R3 -> R3 + R1, we get \Delta = |a b c; a-b c-a c-a; a+b+c a+b+c a+b+c|. Factoring out (a+b+c) from R3, we get \Delta = (a+b+c) * |a b c; a-b c-a c-a; 1 1 1|. Since a, b, c are roots of x^3 - 5x^2 + 3x - 1 = 0, a+b+c = 5. Further simplification leads to the value 80.
Correct Answer: 80