CUET UG 2025 — Mathematics Algebra

Matrix $A$A is $2 \times 4$2×4 (2 rows, 4 columns)

Matrix $B$B is $4 \times 2$4×2 (4 rows, 2 columns)

When multiplying $AB$AB:

$(2 \times 4) \times (4 \times 2) = 2 \times 2$(2×4)×(4×2)=2×2 matrix

Since $AB$AB gives a $2 \times 2$2×2 matrix, which is a square matrix, the determinant $|AB|$∣AB∣ exists.

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Matrix $A$A is $2 \times 4$2×4, which is not square, so $|A|$∣A∣ does not exist.

Matrix $B$B is $4 \times 2$4×2, which is not square, so $|B|$∣B∣ does not exist.

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For any $n \times n$n×n matrix $M$M and scalar $k$k:

$$|kM| = k^n|M|$$∣kM∣=kn∣M∣

where $n$n is the size of the square matrix.

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$AB$AB is a $2 \times 2$2×2 matrix, so $n = 2$n=2

Scalar $k = 3$k=3

$$|3AB| = 3^2|AB|$$∣3AB∣=32∣AB∣

$$|3AB| = 9|AB|$$∣3AB∣=9∣AB∣

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The power depends on the final matrix size after multiplication, not the intermediate dimensions.

Therefore, $|3AB| = 9|AB| = 3^2|AB|$∣3AB∣=9∣AB∣=32∣AB∣