Understanding Adjugate Matrix and Determinants

The question asks for the determinant of the adjugate matrix, denoted as $|adj A|$, given a non-singular matrix $A$ of order 3 with a determinant $|A| = 15$.

Key Relationship: Determinant of Adjugate Matrix

For any square matrix $A$ of order $n$, there is a fundamental relationship between its determinant $|A|$ and the determinant of its adjugate matrix $|adj A|$. This relationship is given by the formula:

$|adj A| = |A|^n-1$

Applying the Formula to the Problem

In this specific problem, we are provided with the following information:

• The order of the matrix $A$ is $n = 3$.
• The matrix $A$ is non-singular, which means its determinant is not zero.
• The determinant of matrix $A$ is $|A| = 15$.

Now, we substitute these values into the formula:

$|adj A| = |A|^n-1$

$|adj A| = 15^3-1$

$|adj A| = 15²$

Calculation

To find the final value, we calculate $15²$:

$15² = 15 × 15 = 225$

Conclusion

Therefore, the determinant of the adjugate matrix $|adj A|$ is 225.