Question

If the order of matrix A is \(3 \times 3\) and \(|A|=2\), then the value of \(|3adj (|3A|A2)|\) is?

• A. \(3^{10}.2^{21}\)
• B. \(2^{10}.3^{21}\)
• C. \(2^{12}.3^{15}\)
• D. \(3^{12}.2^{15}\)

Answer 1

The Correct Option is B

Given:

• Step 1: Calculate \( |3A| \):

  \( |A| = 2 \)

  \( |3A| = 3^3 \cdot |A| \)

  \( |3A| = 3^3 \cdot 2 = 27 \cdot 2 \)

• Step 2: Adj of \( |3A|A^2 \):

  \( \text{Adj.}(|3A|A^2) = \text{Adj.}\{(3^3 \cdot 2)A^2\} \)

  \( = (2 \cdot 3^3)^2 \cdot (\text{Adj.}A)^2 \)

  \( = 2^2 \cdot 36 \cdot (\text{Adj.}A)^2 \)

• Step 3: Calculate \( |3 \cdot \text{Adj.}(|3A|A^2)| \):

  \( |3 \cdot \text{Adj.}(|3A|A^2)| = |2^2 \cdot 36 \cdot (\text{Adj.}A)^2| \)

  \( = (2^2 \cdot 3^7)^3 \cdot |\text{Adj.}A|^2 \)

  \( = 2^6 \cdot 3^{21} \cdot (|A|^2)^2 \)

  \( = 2^6 \cdot 3^{21} \cdot (2^2)^2 \)

  \( = 2^{10} \cdot 3^{21} \)

• Step 4: Final Value:

  \( |3 \cdot \text{Adj.}(|3A|A^2)| = 2^{10} \cdot 3^{21} \)

Answer 2

The correct option is (B): \(2^{10}.3^{21}\)