Explanation
To solve this problem, we use the fundamental properties of determinants related to matrix multiplication and transposes.
Step 1: Apply determinant properties
We are given ∣A∣=−2. We need to find the value of ∣AAT∣.
Using the property of determinants that ∣XY∣=∣X∣⋅∣Y∣, we can write:
∣AAT∣=∣A∣⋅∣AT∣
Step 2: Use the transpose property
Another important property of determinants is that the determinant of a matrix is equal to the determinant of its transpose:
∣A∣=∣AT∣
Step 3: Calculate the result
Substitute the known value ∣A∣=−2 into the expression:
∣AAT∣=∣A∣⋅∣A∣
∣AAT∣=(−2)⋅(−2)
∣AAT∣=4
Conclusion: The correct option is (d) 4.
