Consider the matrix \[ P=\left(\begin{array}{lll} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right) . \] Let the transpose of a matrix \(X\) be denoted by \(\mathrm{X}^{T}\). Then the number of 3 × 3 invertible matrices Q with integer entries, such that \[ Q^{-1}=Q^{T} \text { and } P Q=Q P, \] is

JEE Advanced · 2025 · 18 May 2025 · Paper 1 · Matrices

Consider the matrix

P = 200020003 .

Let the transpose of a matrix

X

be denoted by

X^{T}

. Then the number of 3 × 3 invertible matrices Q with integer entries, such that

Q^{- 1} = Q^{T} and P Q = QP,

Answer: (C)

Since

Q^{- 1} = Q^{T}

Q

is orthogonal:

Q Q^{T} = I

.

Let

Q = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}

.

From

P Q = QP

2 a_{1} 2 a_{2} 3 a_{3} 2 b_{1} 2 b_{2} 3 b_{3} 2 c_{1} 2 c_{2} 3 c_{3} = 2 a_{1} 2 a_{2} 2 a_{3} 2 b_{1} 2 b_{2} 2 b_{3} 3 c_{1} 3 c_{2} 3 c_{3}

This gives

c_{1} = c_{2} = 0

and

a_{3} = b_{3} = 0

. So

Q = a_{1} a_{2} 0 b_{1} b_{2} 0 00 c_{3}

Orthogonality conditions:

a_{1}^{2} + b_{1}^{2} = 1, a_{2}^{2} + b_{2}^{2} = 1, c_{3}^{2} = 1, a_{1} a_{2} + b_{1} b_{2} = 0

With integer entries:

(a_{1}, b_{1}), (a_{2}, b_{2}) \in {(\pm 1, 0), (0, \pm 1)}

and

c_{3} \in {\pm 1}

.

The orthogonality

a_{1} a_{2} + b_{1} b_{2} = 0

gives 8 valid pairs for the first two columns, and 2 choices for

c_{3}

.

Total:

8 \times 2 = 16

matrices.

JEE Advanced · 2025 · 18 May 2025 · Paper 1 · Single correct option