If $\text{det}\left[\begin{matrix}a & 1 & c\\ b & 1 & d\\ e& 1 & f \end{matrix}\right]= -3$ and $\text{det}\left[\begin{matrix}a & 1 & c\\ b & 2 & d\\ e& 3 & f \end{matrix}\right]= 5$ find $\text{det}\left[\begin{matrix}a & -4 & c\\ b & -7 & d\\ e& -10 & f \end{matrix}\right]$.
How do I approach this? The section deals with the effect of row operations on the determinate.
The determinant is linear in every row and in every column. Thus, this problem is equivalent to finding $φ(-4,-7,-10)$ for a linear map $φ \colon ℚ^3 → ℚ$ with $φ(1,1,1) = -3$ and $φ(1,2,3)= 5$. Do you see that – what is $φ$ here? Can you solve this reduced problem?