If the matrices A = [ 1, 1, 2; 1, 3, 4; 1, - 1, 3 ],
B = adjA and C = 3A, then (| adjB |)/(| C |) is equal to :

Solution

Maths sir · 4 steps

Find |A| once, then let the adjoint and scaling rules do all the work.

1. Find det A. Expand along the first row: |A| = 1(9+4) - 1(3-4) + 2(-1-3) = 13 + 1 - 8 = 6.
2. Adjoint rule. For a 3 × 3 matrix, |adj M| = |M|². So |B| = |A|², and applying the rule again gives |adj B| = |B|² = |A|⁴ = 6⁴ = 1296.
3. Scaling rule. Multiplying a 3 × 3 matrix by 3 multiplies every row by 3, so |3A| = 3³|A| = 27 × 6 = 162.

   (|adj B|)/|C| = |A|⁴/27|A| = |A|³/27 = 216/27 = 8

4. So. The ratio is 8, option (a).

The idea to keep. For an n × n matrix, remember |adj A| = |A|ⁿ⁻¹ and |kA| = kⁿ|A|. Chain them and you never need to write out a single adjoint.

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