JEE Advanced2026MathematicsMatricesActual
Consider the matrix M = bmatrix 2 & -1 1 & 0 bmatrix . Let p, q, r, s, a, b, c and d be integers such that M²⁶ = bmatrix p & q r & s bmatrix and _ k=1 ²⁶ M^k = bmatrix a & b c & d bmatrix . Then which of the following statements is (are) TRUE?
Options
- AThere exists a 2 2 invertible matrix N with real entries such that MN = N bmatrix 1 & 1 0 & 1 bmatrix
- BThe value of a is 378
- CFor any two given integers m and n , there exist unique integers x and y such that px + qy = m and rx + sy = n
- DFor each positive real number t , the system of linear equations (a + t)x + by = 1 and cx + (d + t)y = -1 has
Correct answer
A. There exists a 2 2 invertible matrix N with real entries such that MN = N bmatrix 1 & 1 0 & 1 bmatrix
Step-by-step solution
The characteristic equation of M is given by (M - I) = 0 : vmatrix 2 - & -1 1 & - vmatrix = ^2 - 2 + 1 = ( - 1)^2 = 0 The only eigenvalue is = 1 with algebraic multiplicity 2 . For the eigenvector, (M - I)X = 0 bmatrix 1 & -1 1 & -1 bmatrix bmatrix x y bmatrix = bmatrix 0 0 bmatrix x = y . Since there is only one linearly independent eigenvector, M is not diagonalizable but can be reduced to its Jordan canonical form J = bmatrix 1 & 1 0 & 1 bmatrix . Thus, there exists an invertible matrix N such that M = N J N⁻¹ M