Exams › JEE Advanced › Maths

Let P = [[1,0,0],[9,1,0],[27,9,1]] and Q = [q_ij] (3x3) be two matrices satisfying P⁵ - Q = I (where I is the 3x3 identity matrix). Find the value of (q21 + q31) / q32.

1. 22
2. 33
3. 44
4. 55

Correct answer: 22

Solution

Decomposing P = I + N with N nilpotent (N³ = 0) gives P⁵ = I + 5N + 10N², so Q = P⁵ - I = 5N + 10N². Reading off entries q21 = 45, q31 = 945, q32 = 45 gives (45 + 945)/45 = 22.

Related JEE Advanced Maths questions

• Given that A and B are symmetric matrices and they commute (AB = BA), what type of matrix is A^T B?
• Given two matrices A and B satisfying AB = B and BA = A, what is the value of A² + B²?
• Consider the matrix P = [1, 0, 0; 4, 1, 0; 16, 4, 1] and the identity matrix I of size 3. If a matrix Q = [q_(ij)] satisfies P⁵⁰ - Q = I, what is the value of (q₃₁ + q₃₂)/(q₂₁) ?
• Which of the following matrices cannot be expressed as the square of a 3 × 3 matrix with real elements?
• What is the total number of 3 × 3 matrices M, whose elements are chosen from {0, 1, 2}, such that the sum of the diagonal elements of MᵀM equals 5?
• Given the matrix M = [[5/2, 3/2], [−3/2, −1/2]], which of the following represents the value of M raised to the power 2022?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →