<p>If det(A)=5, then det((\(A\)^T)^{-1}) equals:</p>
Step-by-Step Solution
Key Concept: Correctly applying the determinant properties for matrix transpose, i.e., \(\det(A^T) = \det(A)\), and for matrix inverse, i.e., \(\det(A^{-1}) = \frac{1}{\det(A)}\), in the correct order or combination.
If \(\det(A) = 5\), then \(\det((A^T)^{-1}) = \mathbf{\frac{1}{5}}\) (or 0.2)
This is calculated using two basic properties of determinants:Transpose property: \(\det(A^T) = \det(A)\)Inverse property: \(\det(A^{-1}) = \frac{1}{\det(A)}\)Applying these:\(\det((A^T)^{-1}) = \frac{1}{\det(A^T)} = \frac{1}{\det(A)} = \frac{1}{5}\)
<div class="key-concept"><strong>Key Concept:</strong> Correctly applying the determinant properties for matrix transpose, i.e., \(\det(A^T) = \det(A)\), and for matrix inverse, i.e., \(\det(A^{-1}) = \frac{1}{\det(A)}\), in the correct order or combination.</div>
<div class="trap-box"><strong>Trap:</strong> Confusing the property \(\det(A^{-1}) = \frac{1}{\det(A)}\) with \(\det(A^{-1}) = \det(A)\), which would lead to the incorrect answer of 5.</div>
Correct Answer: 1