CUET UG 2025 — Mathematics Algebra

We need to find the determinant of $A$A raised to the power 2025, written as $|A^{2025}|$∣A2025∣.

For any matrix $M$M and positive integer $n$n:

$$|M^n| = |M|^n$$∣Mn∣=∣M∣n

Therefore:

$$|A^{2025}| = |A|^{2025}$$∣A2025∣=∣A∣2025

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The given matrix is:

$$A = \begin{bmatrix} 0 & 1 & -3 \\ -1 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$$A=​0−13​10−5​−350​​

Observe that:

• All diagonal elements are 0
• The element at position (1,2) is 1, and at position (2,1) is -1
• The element at position (1,3) is -3, and at position (3,1) is 3
• The element at position (2,3) is 5, and at position (3,2) is -5

Each pair of opposite elements have opposite signs. This is a skew-symmetric matrix where $A^T = -A$AT=−A.

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For any skew-symmetric matrix of odd order (like 3×3), the determinant is always 0.

Since matrix $A$A is 3×3 (odd order) and skew-symmetric:

$$|A| = 0$$∣A∣=0

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Using the property $|A^{2025}| = |A|^{2025}$∣A2025∣=∣A∣2025:

$$|A^{2025}| = |A|^{2025}$$∣A2025∣=∣A∣2025

$$= 0^{2025}$$=02025

$$= 0$$=0

Therefore, the value of $|A^{2025}| = 0$∣A2025∣=0.