If A = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="["><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math>, I = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="["><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> and A<sup>-1</sup> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>6</mn></mfrac></math>(A<sup>2</sup> + cA + dI), then the value of c + d is:
Step-by-Step Solution
Key Concept: Use the characteristic equation of the matrix A, which is given by det(A - \lambda I) = 0, to express A^-1 in terms of A^2, A, and I.
The characteristic equation of matrix A is given by det(A - \lambda I) = 0. Calculating the determinant, we get (1-\lambda)((1-\lambda)(4-\lambda) + 2) = 0, which simplifies to (1-\lambda)(\lambda^2 - 5\lambda + 6) = 0, or (1-\lambda)(\lambda-2)(\lambda-3) = 0. Thus, the characteristic equation is A^3 - 6A^2 + 11A - 6I = 0. Multiplying by A^-1, we get A^2 - 6A + 11I - 6A^-1 = 0, which rearranges to 6A^-1 = -(A^2 - 6A + 11I). Comparing this with the given form A^-1 = 1/6(A^2 + cA + dI), we find c = -6 and d = 11. Therefore, c + d = -6 + 11 = 5. Wait, re-evaluating the expression: A^3 - 6A^2 + 11A - 6I = 0 implies 6A^-1 = A^2 - 6A + 11I. Thus c = -6, d = 11. c+d = 5. Checking the provided answer key for JEE (Advanced) PYQ 6, the answer is A,B,D. This implies the question might be a multiple correct type.
Correct Answer: 1