# Find Det (C) knowing Det(A) and Det(B)

mmmboh
Hi, right now I have two 3x3 matrices with letter entries, I know the determinant of A and the determinant of B, and I am given a matrix C for which I have to find the determinant for. Right now I am expressing matrix C as a combination of A and B, but what happens to the determinant when I add matrices A and B together? Homework Helper
One of the rows of C is a linear combination of two vectors. det is linear in the row entries, right?

mmmboh
Well all the rows in C are a linear combination of vectors in A and B.

Homework Helper
Well all the rows in C are a linear combination of vectors in A and B.

I'm referring to the second row being a sum of two vectors.

mmmboh
Ok, that's true, but how does that help me in finding the new determinant? I still have to know what happens when I add the matrices together, unless you mean something else?

Homework Helper
I mean that if the letters A, B, C, and D are row vectors then det(A,B+D,C)=det(A,B,C)+det(A,D,C). det is linear in the row vectors.

mmmboh
Ah I didn't know you can do that :S...thanks!

mmmboh
But wait I don't understand how that works...why do the B and D add but nothing else...isn't this sort of like saying det(A+B)=det(A)+det(B) which isn't true?