Suppose C and D and 5x5 matrices and det(C)=5, det(D)=6. Compute the determinant of the matrix 3C^T (4CD)^-1 C^2. Justify your answer by stating explicitly which calculation rules for the determinant you use.
To compute the determinant of the given matrix, we can break it down step by step. Let's start by analyzing each term individually:
1. 3C^T: This represents the transpose of matrix C multiplied by the scalar 3. The determinant of a transpose is the same as the determinant of the original matrix, so det(3C^T) = det(3C) = 3^5 * det(C) = 3^5 * 5 = 243 * 5 = 1215.
2. (4CD)^-1: This is the inverse of the matrix 4CD. The determinant of an inverse matrix is the reciprocal of the determinant of the original matrix, so det((4CD)^-1) = 1 / det(4CD) = 1 / (4^5 * 6) = 1 / 4096 * 6 = 1 / 24576.
3. C^2: This denotes the square of matrix C, which means multiplying it by itself. The determinant of a square matrix raised to a power is the determinant raised to the same power, so det(C^2) = (det(C))^2 = 5^2 = 25.
Now, let's put all the terms together:
det(3C^T (4CD)^-1 C^2) = det(3C^T) * det((4CD)^-1) * det(C^2)
= 1215 * (1 / 24576) * 25
= 1215 / 24576 * 25
= 3 * 5 / 64 * 25
= 15 / 16 * 25
= 375 / 16
= 23.4375
Therefore, the determinant of the given matrix is 23.4375.