Suppose C and D are 2 × 2 matrices with det(C) = −3 and det(D) = 2. Find the determinant of the matrices 8C^3D2 and 5 C^-1
To find the determinant of the matrices 8C^3D^2 and 5C^(-1), we need to understand how matrix operations affect determinants.
For the matrix 8C^3D^2:
1. The determinant of a scalar multiple of a matrix is equal to the scalar multiplied by the determinant of the original matrix. Therefore, the determinant of 8C^3D^2 is (8^2) * det(C)^3 * det(D)^2.
For the matrix 5C^(-1):
1. The determinant of the inverse of a matrix is equal to the reciprocal of the determinant of the original matrix. Therefore, the determinant of 5C^(-1) is 5 / det(C).
Now that we know how to calculate the determinants of these matrices, we can substitute the given determinants into the formulas:
For 8C^3D^2:
Determinant = (8^2) * det(C)^3 * det(D)^2
= 64 * (-3)^3 * 2^2
= 64 * (-27) * 4
= -6912
For 5C^(-1):
Determinant = 5 / det(C)
= 5 / (-3)
= -5/3
Therefore, the determinant of 8C^3D^2 is -6912, and the determinant of 5C^(-1) is -5/3.