Explanation
To solve this, we use the properties of determinants and the adjoint of a matrix.
Step 1: Calculate the determinant of M
Matrix M is a diagonal matrix:
M=200amp;0amp;2amp;0amp;0amp;0amp;2
The determinant of a diagonal matrix is the product of its diagonal elements:
∣M∣=2×2×2=8
Step 2: Use the property of the adjoint determinant
For an n×n matrix A, the property for the determinant of its adjoint is:
∣adjA∣=∣A∣n−1
Here, M is a 3×3 matrix (n=3):
∣adjM∣=∣M∣3−1=∣M∣2
∣adjM∣=82=64
Step 3: Calculate the final expression
We need to find ∣M∣⋅∣adjM∣:
∣M∣⋅∣adjM∣=8⋅64=512
Conclusion: The correct option is (d) 512.
