Exams › JEE Advanced › Maths

Let A be a 2x2 real matrix with |A| != 0 satisfying |A + |A| * adj(A)| = 0. Define B = A - |A| * adj(A). Find the value of tr(A) + |adj(AB)|.

1. 1
2. 2
3. 3
4. 4

Correct answer: 4

Solution

The determinant condition forces the eigenvalues of A + |A|*adj(A) to be zero. For a 2x2 real matrix with complex eigenvalues lambda and conj(lambda), working through the algebra gives |A|=1 and eigenvalues +i, -i, so tr(A)=0. Then |B| = |A - adj(A)| = 4, and |adj(AB)| = |A||B| = 4. Total = 0 + 4 = 4.

Related JEE Advanced Maths questions

• What is the nature of the solution for the equations x + y + z = 3, 2x + y + 2z = 5, and x − y + 3z = 3?
• What is the nature of the solutions for the equations x + y + z = 3, 2x + 2y + 2z = 7, and x − y + 3z = 3?
• Consider the matrix M = [0 1 a; 1 2 3; 3 b 1] and its adjugate adj M = [-1 1 -1; 8 -6 2; -5 3 -1], where a and b are real numbers. Which of the following statements is/are true?
• Given that the 3x3 determinant |x, 2, x; x², x, 6; x, 1, x| equals Ax⁴ + Bx³ + Cx² + Dx + E, find the sum of the digits of the square of (5A + 4B + 3C + 2D + E).
• The determinant with rows (x², (y+z)², yz), (y², (x+z)², zx), (z², (x+y)², xy) is divisible by which of the following?
• Consider the determinant equation: det([[a1 + b1*x, a1*x + b1, c1], [a2 + b2*x, a2*x + b2, c2], [a3 + b3*x, a3*x + b3, c3]]) = 0. Which of the following are possible conditions that guarantee this holds for all choices of ai, bi, ci?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →