Determinant Calculation for Matrix Transpose

The problem requires finding the determinant of the product \(AA^T\), where \(A\) is a square matrix with a known determinant \(|A| = -2\). We need to calculate \(|AA^T|\).

Key Determinant Properties Utilized

Two essential properties of determinants are used here:

• Product Rule: The determinant of a product of two square matrices is the product of their determinants. Mathematically, for square matrices \(X\) and \(Y\), \(|XY| = |X| \cdot |Y|\).
• Transpose Rule: The determinant of the transpose of a square matrix is equal to the determinant of the original matrix. Mathematically, for a square matrix \(X\), \(|X^T| = |X|\).

Calculation Steps

1. Apply the Product Rule to \(|AA^T|\): \(|AA^T| = |A| \cdot |A^T|\)
2. Apply the Transpose Rule (\(|A^T| = |A|\)) to the expression derived in step 1: \(|AA^T| = |A| \cdot |A| = (|A|)^2\)
3. Substitute the given value \(|A| = -2\) into the equation: \(|AA^T| = (-2)^2\)
4. Compute the final result: \(|AA^T| = 4\)

Thus, the determinant \(|AA^T|\) is 4.