Question

Let $P$ be a $5 \times 5$ matrix such that $\det(P) = 2$. If $Q$ is the cofactor matrix of $P$, then find $\det(Q)$.

Hint:

For an $n \times n$ matrix: $\det(\text{adj}(A)) = (\det A)^{n-1}$. Transpose does not change determinant.

Answer 1

Correct Answer: 16

Step 1: Understanding the Topic
This question deals with the relationship between a matrix, its determinant, its cofactor matrix, and its adjugate (or adjoint) matrix. We need to find the determinant of the cofactor matrix using known properties of these related matrices.
Step 2: Key Formula or Approach
We will use two key properties from linear algebra:

The relationship between the cofactor matrix ($Q$) and the adjugate matrix ($\text{adj}(P)$): $\text{adj}(P) = Q^T$.
A crucial identity relating the determinant of the adjugate matrix to the determinant of the original matrix: For an $n \times n$ matrix $P$, $\det(\text{adj}(P)) = (\det P)^{n-1}$.
Step 3: Detailed Calculation
A. Relate the determinants of Q and adj(P):
We know that the determinant of a matrix is equal to the determinant of its transpose. \[ \det(Q) = \det(Q^T) \] Since $Q^T = \text{adj}(P)$, we have: \[ \det(Q) = \det(\text{ajd}(P)) \] B. Apply the determinant of the adjugate formula:
The given matrix $P$ is a $5 \times 5$ matrix, so $n=5$. The formula becomes: \[ \det(\text{adj}(P)) = (\det P)^{5-1} = (\det P)^4 \] C. Substitute the given value for det(P):
We are given that $\det(P) = 2$. \[ \det(Q) = (\det P)^4 = (2)^4 = 16 \] Step 4: Final Answer
The determinant of the cofactor matrix $Q$ is 16. \[ \boxed{16} \]