Some time ago I came across this apparently quite obscure formula that expands the determinant of a sum of two matrices that I had put on my notes (assuming that I made no errors in my writing):
$$\det(A+B)=\det(A)+\det(B)+\text{Tr}(\text{adj}(A)B)$$
Where $\text{adj}()$ denotes the adjugate of the matrix. I cannot seem to find any mention of this formula online. Does anyone know of the name (and maybe a proof) of it? Furthermore, is there any more info on it, like conditions that $A$ and $B$ must obey for it to hold?
Best Answer
For $2 \times 2$ matrices, the following holds
$$\det (\mathrm I_2 + \mathrm M) = 1 + \det(\mathrm M) + \mbox{tr}(\mathrm M)$$
If $\rm A, B$ are $2 \times 2$ matrices, and temporarily assuming that $\rm A$ is invertible, then
$$\begin{array}{rl} \det (\mathrm A + \mathrm B) &= \det \left( \mathrm A \left( \mathrm I_2 + \mathrm A^{-1} \mathrm B \right) \right)\\ &= \det (\mathrm A) \cdot \det \left(\mathrm I_2 + \mathrm A^{-1} \mathrm B \right)\\ &= \det (\mathrm A) \cdot \left( 1 + \det(\mathrm A^{-1} \mathrm B) + \mbox{tr}(\mathrm A^{-1} \mathrm B) \right)\\ &= \det (\mathrm A) + \det (\mathrm A) \cdot \det(\mathrm A^{-1} \mathrm B) + \mbox{tr} \left( \det (\mathrm A) \, \mathrm A^{-1} \mathrm B \right)\\ &= \det (\mathrm A) + \det (\mathrm B) + \mbox{tr} \left( \mbox{adj}(\mathrm A) \mathrm B \right)\end{array}$$