Relation between determinant of the cofactor matrix and the matrix itself?

Question

If the cofactor matrix of A is $$\begin{bmatrix}1 & 2 & 3 \\ 0 & -2 & 4 \\ 0 & 0 & -2\end{bmatrix}$$

How can I find the determinant of $A$?

score 4 · Accepted Answer · answered Oct 03 '15 at 15:10

4

HINT: Let $C$ be the cofactor matrix, then $$C^TA = \det(A)I$$ Take the determinant of both sides of this to get $$\det(C^TA) = \det(\det(A)I) \\ \det(C^T)\det(A) = \det(A)^3\det(I) = \det(A)^3$$

answered Oct 03 '15 at 15:10

so the determinant of C transpose is equal to det(A)^2? – user276520 Oct 03 '15 at 15:17
As long as $A$ is a $3\times 3$ matrix -- yep. – Oct 03 '15 at 15:17

score 1 · Answer 2 · answered Oct 03 '15 at 15:13

1

You can use the fact that $BA=||A||I$ where $B$ is the adjoint of $A$.

answered Oct 03 '15 at 15:13

Ash

669

IMO, it's better to call it the adjugate of $A$ and reserve the word adjoint for the (conjugate) transpose. – Oct 03 '15 at 15:15

score 1 · Answer 3 · answered Oct 03 '15 at 15:38

Actually there are two solutions:

$$A_1=\begin{bmatrix}-2 & 0 & 0 \\ -2 & 1 & 0 \\ -7 & 2 & 1\end{bmatrix}$$ $$A_2=\begin{bmatrix}2 & 0 & 0 \\ 2 & -1 & 0 \\ 7 & -2 & -1\end{bmatrix}$$

You can find these by computing $B={C^{T}}^{-1}$ and solving $A=kB$ and $k=\det(A)$.

$\det(A_1)=-2$ and $\det(A_2)=2$.

Relation between determinant of the cofactor matrix and the matrix itself?

3 Answers3