JEE Main 2026 Jan 22 Shift 1, Mathematics Q13: Algebra Of Determinants

JEE Main2026Jan 22, Shift 1Mathematics

If $A = [2335]$ , then the determinant of the matrix $(A^{2025} - 3 A^{2024} + A^{2023})$ is

Solution

Factor: $A^{2025} - 3 A^{2024} + A^{2023} = A^{2023} (A^{2} - 3 A + I)$ .

$∣ A ∣ = 10 - 9 = 1$ , so $∣ A^{2023} ∣ = 1$ . Hence $det (A^{2023} (A^{2} - 3 A + I)) = ∣ A^{2} - 3 A + I ∣$ .

$A^{2} = [13212134]$ .

$A^{2} - 3 A + I = [13 - 6 + 121 - 9 21 - 934 - 15 + 1] = [8121220]$ .

Determinant $= 8 \cdot 20 - 12 \cdot 12 = 160 - 144 = 16$ .

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