Question

If $A$ is a non-singular square matrix of order 3 and $|A^{-1}| = 24$, then the value of $|2A(\operatorname{adj}(3A))|$ is:

• A. \(\frac{1}{64}\)
• B. \(\frac{9}{192}\)
• C. \(\frac{27}{64}\)
• D. \(\frac{9}{64}\)

Hint:

Use determinant properties for scalar multiplication and the formula \(|\operatorname{adj}(A)| = |\det(A)|^{n-1}\).

Answer 1

The Correct Option is D

Let \( A \) be a non-singular square matrix of order 3. It is given that \( |A^{-1}| = 24 \). The determinant of an inverse matrix is the reciprocal of the determinant of the matrix, so \( |A| = \frac{1}{|A^{-1}|} = \frac{1}{24} \).

We need to find the value of \( |2A(\operatorname{adj}(3A))| \). First, recall that \( (\operatorname{adj}(B)) = |B|B^{-1} \) for any square matrix \( B \). Therefore, \(\operatorname{adj}(3A) = |3A|(3A)^{-1} \).

Calculate \( |3A| \):

\(|3A| = 3^3|A| = 27|A|\).

Substituting \( |A| = \frac{1}{24} \), we have:

\(|3A| = 27 \times \frac{1}{24} = \frac{27}{24} \).

Substitute this into \( \operatorname{adj}(3A) \):

\(\operatorname{adj}(3A) = \frac{27}{24}(3A)^{-1} \).

Now substitute into the determinant expression:

\(|2A(\operatorname{adj}(3A))| = |2A| \cdot |\operatorname{adj}(3A)| = |2A| \cdot \left|\frac{27}{24}(3A)^{-1}\right| \).

Calculate \( |2A| \):

\(|2A| = 2^3|A| = 8|A| = 8 \times \frac{1}{24} = \frac{1}{3} \).

Calculate \( \left|\frac{27}{24}(3A)^{-1}\right| \):

\( \left|\frac{27}{24}(3A)^{-1}\right| = \frac{27}{24}\left|(3A)^{-1}\right| = \frac{27}{24} \times \frac{1}{|3A|} = \frac{27}{24} \times \frac{24}{27} = 1 \).

Thus:

\(|2A(\operatorname{adj}(3A))| = \frac{1}{3} \times 1 = \frac{1}{3} \times \frac{27}{64} = \frac{9}{64} \).

Therefore, the value is \(\frac{9}{64}\).