Animated Solution for Mathematics - Matrices and Determinants: If $A$ is a square matrix of order 3 such that $\det (A)=3$ and $\det (\text{adj}(-4\text{adj}(-3\text{adj}(3\text{adj}((2A{)}^{-1})))))=2^m{3}^n$, then $m+2n$ is equal to:

Unraveling the Nested Determinant

Imagine standing before a grand, intricate clockwork mechanism. Each gear is nested within another, and every turn of the outer gear depends entirely on the subtle, precise movement of the innermost spring.

In linear algebra, specifically when dealing with nested determinants and adjoint matrices, we are essentially reverse-engineering this clockwork. Today, we are going to dismantle a complex matrix expression, not by brute force, but by respecting the elegant laws of determinants.

Understanding the Tools

Before we dive into the abyss of the expression, let us sharpen our tools. We are given a $3\times 3$ matrix $A$ with $\det (A)=3$. Our goal is to find the value of $m+2n$ from the expression $\det (\text{adj}(-4\text{adj}(-3\text{adj}(3\text{adj}((2A{)}^{-1})))))$.

To survive this, we must rely on three fundamental pillars:

1. The Scalar Property: $\det (kA)=k^n\det (A)$, which for $n=3$ becomes $k^3\det (A)$.

2. The Adjoint Property: $\det (\text{adj}(A))=(\det (A){)}^{n-1}$, which for $n=3$ becomes $(\det (A){)}^2$.

3. The Inverse Property: $\det (A^{-1})=\frac{1}{\det (A)}$.

The Innermost Core

We start at the very center: $M_0=(2A{)}^{-1}$. We need the determinant of this core.

Using our properties, we find:

Thus, $\det (M_0)=\frac{1}{24}=2^{-3}\cdot 3^{-1}$. This is the heartbeat of our entire expression, and every subsequent layer will be built upon this value.

Now, we move outward. Let $M_1=3\text{adj}(M_0)$. Applying our scalar and adjoint properties, we get:

As we move to $M_2=-3\text{adj}(M_1)$, we must be careful with the negative sign. Since the order is $n=3$, the scalar $-3$ becomes $(-3{)}^3=-27$.

We are almost there. The next layer is $M_3=-4\text{adj}(M_2)$. Again, the scalar $-4$ becomes $(-4{)}^3=-64=-2^6$.

Finally, we reach the outermost layer $D=\text{adj}(M_3)$. The determinant of the adjoint is the square of the determinant of the matrix:

We were told that $\det (D)=2^m\cdot 3^n$. By comparing our result, we identify $m=-36$ and $n=20$.

The final challenge is to compute $m+2n$:

Look at that! Through systematic application of the laws of linear algebra, a terrifying nested expression collapses into a simple integer. You have successfully navigated the clockwork, proving that complexity is often just a veil.