Let A be a 3 × 3 matrix with det(A) = 4. Let Rᵢ denote the iᵗʰ row of A. If a matrix B is obtained by performing the operation R₂ → 2R₂ + 5R₃ on 2A, then det(B) is equal to :

Solution

Maths sir · 3 steps

Pull out 2³ for the whole matrix, then a single 2 for the row operation. The 5R₃ part changes nothing.

1. Scale the matrix. Multiplying a 3 × 3 matrix by 2 multiplies each of its 3 rows by 2. Each row gives one factor of 2, so det(2A) = 2³ det(A) = 8 × 4 = 32.
2. Split the operation. Think of R₂ → 2R₂ + 5R₃ in two parts. Scaling R₂ by 2 doubles the determinant. Adding 5R₃ to a row does not change it, because adding a multiple of another row never does.

   det(B) = 2 × det(2A) = 2 × 2³ × det(A)

3. So. det(B) = 2 × 32 = 64, which is option (a).

The idea to keep. For an n × n matrix, det(kA) = kⁿ det(A). In Rᵢ → aRᵢ + bRⱼ only the factor a on the row itself matters. The added bRⱼ part leaves the determinant unchanged.

Answer from the NTA final answer key. Solution by PrepWiser.