JEE Advanced2026MathematicsMatricesActual
For real numbers , , , and , consider the matrix M = bmatrix & 1 2 & - 1 2 1 3 & & 1 3 & & bmatrix . Suppose that MM^T = I , where M^T is the transpose of the matrix M , and I is the 3 3 identity matrix. Let u = , i + 1 3 , j + , k , v = 1 2 , i + , j + , k and w = - 1 2 , i + 1 3 , j + , k . Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) The value of ^2 +
Options
- A(P) (5), (Q) (4), (R) (2), (S) (1)
- B(P) (4), (Q) (5), (R) (1), (S) (2)
- C(P) (5), (Q) (3), (R) (2), (S) (1)
- D(P) (5), (Q) (4), (R) (1), (S) (2)
Correct answer
A. (P) (5), (Q) (4), (R) (2), (S) (1)
Step-by-step solution
Since MM^T = I , M is an orthogonal matrix. The columns of an orthogonal matrix form an orthonormal basis for R ^3 . The given vectors u , v , w are exactly the columns of M . Therefore, u , v , w are mutually orthogonal unit vectors. For (P): The rows of M are also orthonormal. The third row is bmatrix & & bmatrix , so ^2 + ^2 + ^2 = 1 . The third column w is a unit vector, so | w |^2 = (- 1 2 )^2 + ( 1 3 )^2 + ^2 = 1 . 1 2 + 1 3 + ^2 = 1 ^2 = 1 6 Substituting ^2 into the row equation gives ^2 + ^2 = 1 - 1 6 = 5 6