cofactor in nonsigular matrix

Question

Let $A = [a_{ij}]$ be a $4 \times 4$ nonsingular matrix with $\det (A) = 5$.

We know that the inverse matrix $A^{-1} = \frac{[C_{ij}]^T}{\det A}$, where $C_{ij}$ is the cofactor of $a_{ij}$.

Find $\det([C_{ij}])$.

Answer 1

You have identified the rank correctly but it is unlikely that that can help you.

Guide:

Try to apply determinant on both sides and use the following identity to solve for $\det(C)$.

• $\det(A^{-1})=\frac1{\det(A)}$
• $\det(C)=\det(C^T)$.
• $\det(kC)=k^n\det(C)$

Your final answer should be of the form of $\det(A)^m=5^m$ where $m$ is a positive integer.