Question

If \( A \) is an invertible matrix, then \( \det(A^{-1}) \) is:

Hint:

The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix: \( \det(A^{-1}) = \frac{1}{\det(A)} \).

Answer 1

Step 1: Understanding the properties of an invertible matrix.
An invertible matrix \( A \) has the property that its determinant is non-zero, i.e., \( \det(A) \neq 0 \). For such matrices, we can also compute the determinant of the inverse matrix, \( A^{-1} \), using a well-known property.
Step 2: The key property of the determinant of the inverse matrix.
The determinant of the inverse matrix \( A^{-1} \) is related to the determinant of the original matrix \( A \). Specifically, the relationship is given by the formula: \[ \det(A^{-1}) = \frac{1}{\det(A)} \] This property follows from the fact that \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix, and \( \det(A \cdot A^{-1}) = \det(I) = 1 \). Using the multiplicative property of determinants, we have: \[ \det(A) \cdot \det(A^{-1}) = 1 \] which leads to: \[ \det(A^{-1}) = \frac{1}{\det(A)} \] Step 3: Conclusion.
Thus, the determinant of \( A^{-1} \) is \( \frac{1}{|A|} \), where \( |A| \) denotes the determinant of matrix \( A \). Therefore, the correct answer is \( \frac{1}{|A|} \).