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: Use relation between adjoint and determinant.
For any $n \times n$ matrix $A$:
\[ \text{adj}(A) = (\text{cofactor matrix})^T. \]
And the important identity:
\[ A \cdot \text{adj}(A) = \det(A) I. \]
Also, \[ \det(\text{adj}(A)) = (\det A)^{n-1}. \]
Step 2: Apply formula for $n=5$.
Since $P$ is $5 \times 5$,
\[ \det(\text{adj}(P)) = (\det P)^{5-1}. \]
\[ = (\det P)^4. \]
Step 3: Substitute given value.
\[ \det(P) = 2. \]
\[ \det(\text{adj}(P)) = 2^4 = 16. \]
Step 4: Relation with cofactor matrix.
Cofactor matrix and adjoint differ only by transpose.
Since determinant of a matrix equals determinant of its transpose:
\[ \det(Q) = \det(\text{adj}(P)). \]
\[ = 16. \]
Final Answer:
\[ \boxed{16}. \]