Question

If \( A = \begin{pmatrix} 2024 & 2021 \\ 2023 & 2022 \end{pmatrix} \), then the value of \[ \left| A^{2024} - A^{2023} \right| \] is

• A. 2024
• B. 1
• C. 0
• D. 2023

Hint:

Use determinant properties: \( |A^n| = |A|^n \) and \( |AB| = |A||B| \) to simplify matrix power determinant problems.

Answer 1

The Correct Option is C

Using the property of determinants: \[ |A^n| = |A|^n \] So, \[ \left| A^{2024} - A^{2023} \right| = \left| A^{2023} (A - I) \right| \] Then, by determinant multiplication property: \[ = |A^{2023}| \times |A - I| \] Now, \[ |A| = (2024 \times 2022) - (2021 \times 2023) \] Simplifying: Left diagonal product: \(2024 \times 2022 = 4093632\) Right diagonal product: \(2021 \times 2023 = 4093632\) So, \[ |A| = 4093632 - 4093632 = 0 \] Therefore, \[ |A^{2023}| = 0 \] And finally, \[ \left| A^{2024} - A^{2023} \right| = 0 \times |A - I| = 0 \]