If A is a 3 × 3 matrix and |A| = 2, then |3 adj (|3A|A²)| is equal to :

Solution

Maths sir · 4 steps

Work from the inside out: turn |3A| into a number first, then peel off one layer at a time.

1. Inner determinant. For a 3 × 3 matrix, |kA| = k³|A|. So |3A| = 27 × 2 = 54, which is just a number.
2. Name the matrix. Let B = 54A². Then |B| = 54³ |A|² = 54³ × 4.

   |3 adj B| = 3³ |adj B| = 3³ |B|² = 3³ × 54⁶ × 16

3. Prime factors. Use |adj B| = |B|ⁿ⁻¹ = |B|² for n = 3. Write 54 = 2 × 3³, so 54⁶ = 2⁶ × 3¹⁸. The product is 3³ × 2⁶ × 3¹⁸ × 2⁴ = 2¹⁰ × 3²¹.
4. So. 2¹⁰ × 3²¹ = (2¹⁰ × 3¹⁰) × 3¹¹ = 3¹¹ · 6¹⁰, which is option (b).

The idea to keep. For an n × n matrix, remember three rules: |kA| = kⁿ|A|, |Aᵐ| = |A|ᵐ and |adj A| = |A|ⁿ⁻¹. Reduce every number to primes before matching options.

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