CUET UG 2025 — Mathematics Algebra

Given $A$A is a 3×3 square matrix (order 3) and $|A| = 5$∣A∣=5.

The expression $|-AA^T|$∣−AAT∣ represents the determinant of $(-A)$(−A) multiplied by the transpose of matrix $A$A.

$$|-AA^T| = |(-A) \times A^T|$$∣−AAT∣=∣(−A)×AT∣

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When two matrices multiply, their determinants also multiply:

$$|(-A) \times A^T| = |{-A}| \times |A^T|$$∣(−A)×AT∣=∣−A∣×∣AT∣

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When a matrix is multiplied by $-1$−1, the determinant is affected by $(-1)^n$(−1)n, where $n$n is the order of the matrix.

$$|{-A}| = (-1)^n \times |A|$$∣−A∣=(−1)n×∣A∣

Since the matrix has order 3:

$$|{-A}| = (-1)^3 \times 5$$∣−A∣=(−1)3×5

$$|{-A}| = -1 \times 5$$∣−A∣=−1×5

$$|{-A}| = -5$$∣−A∣=−5

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The determinant of a matrix equals the determinant of its transpose:

$$|A^T| = |A| = 5$$∣AT∣=∣A∣=5

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$$|-AA^T| = |{-A}| \times |A^T|$$∣−AAT∣=∣−A∣×∣AT∣

$$|-AA^T| = (-5) \times 5$$∣−AAT∣=(−5)×5

$$|-AA^T| = -25$$∣−AAT∣=−25

Therefore, the value of $|-AA^T| = -25$∣−AAT∣=−25.