Matrix Simplification and Determinant Calculation

First, simplify the given matrix $A$ using row operations to find its determinant.

Given matrix:

$ A = \begin{bmatrix} 2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q \end{bmatrix} $

Apply row operations:

• $R_2 \leftarrow R_2 - 2R_1$
• $R_3 \leftarrow R_3 - 3R_1$

The matrix transforms into:

$ \begin{bmatrix} 2 & 2+p & 2+p+q \\ 0 & 2 & 4+p \\ 0 & 6 & 14+3p \end{bmatrix} $

Now, calculate the determinant of $A$ using the cofactor expansion along the first column:

$ \det(A) = 2 \times \begin{vmatrix} 2 & 4+p \\ 6 & 14+3p \end{vmatrix} $ $ \det(A) = 2 \times [2(14+3p) - 6(4+p)] $ $ \det(A) = 2 \times [28 + 6p - 24 - 6p] $ $ \det(A) = 2 \times [4] = 8 $

Properties of Adjoint and Determinant

We need to evaluate $\det(\text{adj}(\text{adj}(3A)))$. Let the matrix size be $n=3$. We use the following properties for an $n \times n$ matrix $X$:

• $\det(kA) = k^n \det(A)$
• $\text{adj}(\text{adj}(X)) = (\det(X))^{n-2} X$ (for $n \ge 3$)

Let $X = 3A$. Using the property $\text{adj}(\text{adj}(X)) = (\det(X))^{n-2} X$, we get:

$ \text{adj}(\text{adj}(3A)) = (\det(3A))^{3-2} (3A) = (\det(3A)) (3A) $

Now, find the determinant of this expression:

$ \det(\text{adj}(\text{adj}(3A))) = \det((\det(3A)) (3A)) $

Using $\det(kA) = k^n \det(A)$, where $k = \det(3A)$ and the matrix is $3A$ (size $n=3$):

$ \det(\text{adj}(\text{adj}(3A))) = (\det(3A))^3 \det(3A) = (\det(3A))^4 $

Calculating $\det(\text{adj}(\text{adj}(3A)))$

First, calculate $\det(3A)$ using $\det(kA) = k^n \det(A)$ with $k=3$ and $n=3$:

$ \det(3A) = 3^3 \det(A) = 27 \times 8 = 216 $

Now substitute this value back into the expression for $\det(\text{adj}(\text{adj}(3A)))$:

$ \det(\text{adj}(\text{adj}(3A))) = (216)^4 $

Express 216 in terms of its prime factors:

$ 216 = 6^3 = (2 \times 3)^3 = 2^3 \times 3^3 $

Therefore:

$ (216)^4 = (2^3 \times 3^3)^4 = (2^3)^4 \times (3^3)^4 = 2^{12} \times 3^{12} $

We are given that $\det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n$. So:

$ 2^{12} \cdot 3^{12} = 2^m \cdot 3^n $

Finding m + n

By comparing the powers of 2 and 3, we find:

• $m = 12$
• $n = 12$

Both $m$ and $n$ are natural numbers, as required.

The value of $m + n$ is:

$ m + n = 12 + 12 = 24 $