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Let's analyze the problem and solve for \( X \) step by step.

The equation given is:

\[
(4B^{-1} X)^{-1} = \left( (3A)^{-1} B^{-1} \right) B A^3
\]

### Step 1: Inverse of both sides
We can start by taking the inverse of both sides to remove the outer inverse on the left-hand side. Using the property that the inverse of an inverse is the original matrix:

\[
4B^{-1} X = \left[ \left( (3A)^{-1} B^{-1} \right) B A^3 \right]^{-1}
\]

### Step 2: Simplify the right-hand side
We simplify the product on the right-hand side first. Notice that \( B^{-1} B = I \) (where \( I \) is the identity matrix), so:

\[
\left( (3A)^{-1} B^{-1} \right) B A^3 = (3A)^{-1} A^3
\]

Now, simplify \( (3A)^{-1} A^3 \). This can be broken down further by using the property \( (AB)^{-1} = B^{-1} A^{-1} \), and knowing that \( A^{-1} A = I \):

\[
(3A)^{-1} A^3 = \frac{1}{3} A^{-1} A^3 = \frac{1}{3} A^2
\]

Thus, the equation becomes:

\[
4B^{-1} X = \left( \frac{1}{3} A^2 \right)^{-1}
\]

### Step 3: Inverse of the right-hand side
Now, take the inverse of \( \frac{1}{3} A^2 \). The inverse of a scalar multiplied by a matrix is given by \( \left( c A \right)^{-1} = \frac{1}{c} A^{-1} \), so:

\[
\left( \frac{1}{3} A^2 \right)^{-1} = 3 A^{-2}
\]

Now the equation is:

\[
4B^{-1} X = 3 A^{-2}
\]

### Step 4: Solve for \( X \)
Multiply both sides of the equation by \( B \) to eliminate \( B^{-1} \) from the left-hand side:

\[
4X = 3 B A^{-2}
\]

Finally, divide both sides by 4:

\[
X = \frac{3}{4} B A^{-2}
\]

### Final Answer:
\[
X = \frac{3}{4} B A^{-2}
\]

Would you like further clarification on any part of this process or have additional questions?

---

Here are 5 related questions to expand your understanding:

1. What is the significance of the inverse of a matrix in solving matrix equations?
2. How can you verify if a matrix is invertible?
3. What properties of matrix multiplication and inversion are crucial for solving this problem?
4. Can you derive \( X \) if the matrices were not square?
5. How does scalar multiplication work with matrices?

**Tip:** Always ensure that the dimensions of matrices are compatible when multiplying or inverting them.

Suppose that A, B, and X are invertible square matrices of the same size. Solve for X in terms of A and B: (4B^{-1}X)^{-1} = ((3A)^{-1}B^{-1})BA^3

This problem involves solving for matrix X in terms of A and B, where A, B, and X are invertible matrices. The solution explores key concepts like matrix inversion and properties of square matrices. Detailed steps include simplification of matrix products and division.

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