Find determinant of cofactor matrix

Question

If have a matrix $A$ that all I know about is its size and what its determinant is? For example a $4\times4$ matrix with a determinant of $3$. How can I find the determinant of the cofactor matrix $cof(A)$.

What I have done is I created a lower triangular matrix where the product of the diagonals will give me $3$. So do I just find the cofactor matrix of the matrix I came up with and find its determinant. Would that be correct?

The cofactor matrix is the transpose of the adjugate matrix, so it suffices to compute the determinant of the latter. See http://math.stackexchange.com/questions/92837/proof-mathrmadj-mathrmadja-mathrmdetan-2-cdot-a-for-a?rq=1 for that. — darij grinberg, Mar 19 '16 at 23:31
It is quite a standard fact that the cofactor matrix has determinant $1/\det A$. — Crostul, Mar 20 '16 at 00:37

score 0 · Answer 1 · answered Dec 17 '22 at 16:14

0

For your $n \times n$ matrix $A$,

is $\det(A) \neq 0$,
is $n > 1$,
do you accept the formulas $A^{-1} = cof(A)^T / \det(A)$,
$\det(A B) = \det(A) \, \det(B)$ for two $n \times n$ martices $A$ and $B$ and
$\det(B^{T}) = \det(B)$ for any matrix $B$ ?

If so, we have

$\det(A) I_n = cof(A)^T A$.

Taking the determinant on both sides, we obtain

$\det(A)^n = \det(cof(A)^T) \, \det(A)$ and so

$\det(cof(A)) = \det(cof(A)^T) = \det(A)^{n-1}$.

answered Dec 17 '22 at 16:14

Neo

93

1

It's worth noting that the result is still true when $\det A=0$ (and even, under suitable conventions, when $n=1$). – blargoner Dec 17 '22 at 17:24

Find determinant of cofactor matrix

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