The problem asks to find the value of $\left|(2A)^{-1} B^T A^2 \right|$, where $A$ and $B$ are square matrices of order 4 with $|A| = -3$ and $|B| = 24$.

Step-by-Step Solution

1. Determinant of Scalar Multiplication of a Matrix: The determinant of a scalar multiple of a matrix $A$ is given by: $|kA| = k^n |A|$ where $k$ is a scalar and $n$ is the order of the matrix. Since $A$ is a 4x4 matrix, $n = 4$. So: $|2A| = 2^4 |A| = 16 \cdot (-3) = -48$

2. Determinant of the Inverse: The determinant of the inverse of a matrix is the reciprocal of the determinant: $|(2A)^{-1}| = \frac{1}{|2A|} = \frac{1}{-48}$

3. Determinant of a Product of Matrices: The determinant of the product of matrices is the product of their determinants: $\left|(2A)^{-1} B^T A^2 \right| = |(2A)^{-1}| \cdot |B^T| \cdot |A^2|$

4. Determinant of $B^T$: The determinant of the transpose of a matrix is the same as the determinant of the original matrix: $|B^T| = |B| = 24$

5. Determinant of $A^2$: The determinant of the square of a matrix is the square of the determinant: $|A^2| = (|A|)^2 = (-3)^2 = 9$

6. Final Calculation: Now, we combine everything: $\left|(2A)^{-1} B^T A^2 \right| = \frac{1}{-48} \cdot 24 \cdot 9 = \frac{1}{-48} \cdot 216 = \frac{216}{-48} = -4.5$

So, the value of $\left|(2A)^{-1} B^T A^2 \right|$ is -4.5.

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Related Questions:

1. How is the determinant of the product of two matrices derived?
2. What is the significance of matrix transposition in determinant calculations?
3. How do scalar multiples affect the determinant of a matrix?
4. Why is the determinant of the inverse matrix the reciprocal of the original determinant?
5. What are the properties of determinants when squaring a matrix?

Tip: Always check if matrix properties like symmetry, diagonal, or transpose affect determinant calculations.