Let A and B be matrices of order 3\times 3. If |A|=\frac{1}{2\sqrt{2}} and |B|=\frac{1}{729} then what is the value of |2B(adj(3A))|?

Explanation

We know |kM| = k^n|M| and |adj(M)| = |M|^{n-1} for an n \times n matrix. Here n=3. |2B(adj(3A))| = 2^3 |B| \cdot |adj(3A)| = 8 |B| |3A|^2. Since |3A| = 3^3 |A| = 27|A|, we get 8 |B| (27|A|)^2 = 8 \cdot \frac{1}{729} \cdot 729 \cdot |A|^2 = 8 \cdot \left(\frac{1}{2\sqrt{2}}\right)^2 = 8 \cdot \frac{1}{8} = 1.

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