Evaluating |adj(2 * adj A)| for a 3x3 Matrix A

This problem requires us to find the specific numerical value of the expression |adj(2 * adj A)|. We are given two crucial pieces of information: the matrix A is of the order of 3 (meaning it is a $3 \times 3$ matrix), and its determinant, |A|, is equal to 15.

Understanding Properties of Adjoint and Determinants

To solve this, we will utilize established properties concerning the determinant and the adjoint of square matrices. These properties allow us to simplify the expression systematically.

The key properties we need are:

• Determinant of an Adjoint Matrix: For any $n \times n$ square matrix $M$, the determinant of its adjoint matrix ($adj M$) is related to the determinant of $M$ by the formula: $$ |adj M| = |M|^{n-1} $$
• Determinant of a Scaled Matrix: For any $n \times n$ square matrix $M$ and any scalar $k$, the determinant of the matrix scaled by $k$ ($kM$) is given by: $$ |kM| = k^n |M| $$

Step-by-Step Calculation of |adj(2 * adj A)|

Let's apply these properties step-by-step to the expression $|adj (2 \cdot adj A)|$, using the given order $n=3$ and determinant $|A|=15$.

1. Expression: We start with the target expression: $$ |adj (2 \cdot adj A)| $$
2. Applying Property 1 (Outer Adjoint): Let's consider the outer $adj$ operation. Here, the matrix is $M = (2 \cdot adj A)$. Since $A$ is a $3 \times 3$ matrix, $adj A$ is also $3 \times 3$, and $2 \cdot adj A$ is still a $3 \times 3$ matrix. So, $n=3$. Applying the property $|adj M| = |M|^{n-1}$: $$ |adj (2 \cdot adj A)| = |2 \cdot adj A|^{3-1} = |2 \cdot adj A|^2 $$
3. Applying Property 2 (Scalar Multiplication): Now, let's evaluate $|2 \cdot adj A|$. Using the property $|kM| = k^n |M|$ with $k=2$, $n=3$, and $M = adj A$: $$ |2 \cdot adj A| = 2^3 |adj A| $$
4. Substituting the Scalar Result: Substitute this result back into the expression from step 2: $$ |adj (2 \cdot adj A)| = (2^3 |adj A|)^2 $$
5. Applying Property 1 (Inner Adjoint): We need the value of $|adj A|$. Using the property $|adj M| = |M|^{n-1}$ with $M=A$ and $n=3$: $$ |adj A| = |A|^{3-1} = |A|^2 $$
6. Substituting |adj A| Value: Substitute the expression for $|adj A|$ from step 5 into the equation from step 4: $$ |adj (2 \cdot adj A)| = (2^3 \cdot |A|^2)^2 $$
7. Using Given Values: Now, substitute the provided values $|A| = 15$ and $n=3$ (implicitly used in the powers): $$ |adj (2 \cdot adj A)| = (2^3 \cdot 15^2)^2 $$
8. Numerical Calculation: Perform the arithmetic:
   • Calculate $2^3 = 8$.
   • Calculate $15^2 = 225$.
   • The expression becomes: $(8 \cdot 225)^2$.
   • Calculate the product inside the parenthesis: $8 \times 225 = 1800$.
   • Calculate the final square: $1800^2 = 3240000$.

Conclusion

By applying the properties of determinants and adjoints, and performing the calculations step-by-step, we find that the value of |adj(2 * adj A)|, given that A is a 3rd order matrix and |A| = 15, is 3240000.