Q37 / 75 · JEE Main 2026 (22 January Shift 1) · Mathematics

If $\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 3 & 5\end{array}\right]$, then the determinant of the matrix $\left(\mathrm{A}^{2025}-3 \mathrm{~A}^{2024}+\mathrm{A}^{2023}\right)$ is

  • A
    12
  • B
    28
  • C
    24
  • Correct answer
    D
    16

Solution

$A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix} \Rightarrow A^2 = \begin{bmatrix} 13 & 21 \\ 21 & 34 \end{bmatrix}$
$|A^{2025} - 3A^{2024} + A^{2023}|$
$= \left| A^{2023} \left( A^2 - 3A + I \right) \right|$
$= |A|^{2023} |A^2 - 3A + I|$
$= 1 \cdot \begin{vmatrix} 8 & 12 \\ 12 & 20 \end{vmatrix} = 160 - 144 = 16$

Hence, the correct answer is option: 'D'.

Relevant Questions