I need to prove that |adj(A)|=|A|^(n-1) for every non singular A in F^nxn, while n>2.
The solution says:
I can understand the first equality - |adj(A)|=||A|*A^(-1)|,
but I can't understand the rest.
Can you please help me?
Thank you!
I need to prove that |adj(A)|=|A|^(n-1) for every non singular A in F^nxn, while n>2.
The solution says:
I can understand the first equality - |adj(A)|=||A|*A^(-1)|,
but I can't understand the rest.
Can you please help me?
Thank you!
Using $\displaystyle det(\alpha A) = \alpha^n det(A)$ , we get:
$\displaystyle det(det(A) * det(A^{-1})) = det(A)^n * det(A^{-1})$
Since $\displaystyle det(A)$ is a scalar in F. Also, $\displaystyle det(A^{-1}) = \frac{1}{det(A)} \Rightarrow det(det(A) * det(A^{-1})) = \frac{det(A)^n}{det(A)} = det(A)^{n-1}$