If A = [ 2 3 3 5 ] , then the determinant of the matrix ( A 2025 - 3 A 2024 + A 2023 ) is [2026]

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Solution

Q.
If $A = [\begin{matrix} 2 & 3 \\ 3 & 5 \end{matrix}],$ then the determinant of the matrix $(A^{2025} - 3 A^{2024} + A^{2023})$ is [2026]

1 16

2 12

3 24

4 28

Ans.
(1)

$A = [\begin{matrix} 2 & 3 \\ 3 & 5 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 13 & 21 \\ 21 & 34 \end{matrix}]$

$| A^{2025} - 3 A^{2024} + A^{2023} |$

$=$ $| A^{2023} (A^{2} - 3 A + I) |$

$= | A |^{2023}$ $| A^{2} - 3 A + I |$

$= 1 \cdot$ $| \begin{matrix} 8 & 12 \\ 12 & 20 \end{matrix} |$ $= 160 - 144 = 16$

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