$\text{det}(A+B)$ is the sum of $\text{det}A$ and a linear combination of minors of $B$.

Question

Let $A$ and $B$ be square matrices. Then $\text{det}(A+B)$ is the sum of $\text{det}A$ and a linear combination of minors of $B$.

I want to show the above statement but get stuck. Anyone can give a hint? Thanks in advance.

Answer 1

Hint

Use induction on the size of the matrix and, for the inductive step

Say $a_{i,j}, b_{i,j}$ are the elements of the matrices $A $ and $B $. Write $\det A+B $ with Laplace formula as

$$\sum_{i=1}^n (a_{1,i} + b_{1,i})\det M_i $$

Where $M_i$ is the submatrix you get by deleting the 1st row and $i $th column from $A+B $. Now rewrite $M_i $ as submatrices from $A, B $, and work your way through the sum, applying the induction hypothesis so that $\det A $ comes up, as well as some linear combination of minors of $B $.