Question:
If
\[
A = \begin{pmatrix} 2 & 3 \\ 3 & 5 \end{pmatrix}
\]
then the value of
\[
|A^{2025} - 3A^{2024} + A^{2023}|
\]
is:
If
\[
A = \begin{pmatrix} 2 & 3 \\ 3 & 5 \end{pmatrix}
\]
then the value of
\[
|A^{2025} - 3A^{2024} + A^{2023}|
\]
is:
Show Hint
When dealing with matrix powers, analyze the properties of the matrix (like eigenvalues) to simplify calculations for large powers.
Updated On: Jan 28, 2026
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Correct Answer: 16
Solution and Explanation
Step 1: Simplifying the expression.
The given matrix is \( A = \begin{pmatrix} 2 & 3
3 & 5 \end{pmatrix} \). We need to compute the expression \( A^{2025} - 3A^{2024} + A^{2023} \). Step 2: Identifying the behavior of powers of matrices.
Since the matrix \( A \) is a 2x2 matrix, we can use the properties of matrix powers and simplify using the fact that the matrix follows a pattern based on its eigenvalues. After applying matrix exponentiation and the given values, we compute the final result. Step 3: Conclusion.
The value of the expression \( |A^{2025} - 3A^{2024} + A^{2023}| \) is 16. Final Answer: \[ \boxed{16} \]
The given matrix is \( A = \begin{pmatrix} 2 & 3
3 & 5 \end{pmatrix} \). We need to compute the expression \( A^{2025} - 3A^{2024} + A^{2023} \). Step 2: Identifying the behavior of powers of matrices.
Since the matrix \( A \) is a 2x2 matrix, we can use the properties of matrix powers and simplify using the fact that the matrix follows a pattern based on its eigenvalues. After applying matrix exponentiation and the given values, we compute the final result. Step 3: Conclusion.
The value of the expression \( |A^{2025} - 3A^{2024} + A^{2023}| \) is 16. Final Answer: \[ \boxed{16} \]
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