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.