Properties Of Determinants

Scaling an entire n by n matrix by a constant k multiplies its determinant by k^n, a property of determinants that JEE Advanced tests routinely. Here A has order n = 3 and det(A) = 5. Multiplying A by 2 scales each of the three rows by 2, and each scaled row contributes a factor of 2 to the determinant, so det(2A) = 2^3 · det(A) = 8 · 5 = 40. Option 10 wrongly uses det(2A) = 2·det(A), valid only for order 1. Option 20 uses 2^2, the wrong power for order 3. Option 80 uses 2^4, an exponent too large. Hence det(2A) = 40. Plausibility check: the exponent on the scalar must equal the matrix order, and for a 3x3 matrix 2^3 = 8 is the only consistent factor, giving 40 and matching the row-by-row scaling argument. The scaling rule det(kA) equals k to the power n times det(A) follows because each of the n rows independently contributes a factor of k to the multilinear determinant. Confusing this with the scalar rule for a single number is a classic trap, so always pairing the exponent with the matrix order keeps the computation aligned with the underlying row-by-row scaling.