Find the determinant $M$ if $M = 3A^2 + AB + B^2$, where $$A = \begin{vmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & -1 & 0 \end{vmatrix}$$ and $$B = \begin{vmatrix} 1/2 & 0 & -1 \\ -1 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$$ without evaluating $A$ and $B$ independently.
My approach was
$$\det(M) = \det( 3A^2 + AB + B^2) = \det(A+B)^2 - \det(AB) + 2\det(A^2) = \det(A+B)^2 + \det(A) \times \det(2A-B)$$
Is the approach correct?
In short, no. As has already been pointed out in comments, determinants are multilinear. In particular, it’s not generally true that $\det(A+B)=\det(A)+\det(B)$. The only property of determinants introduced in the preceding material in the text that hasn’t been used in any of the other exercises is that the determinant of a product is the product of determinants, so my guess is that this problem is meant to provide you with some some practice with that. However tedious the calculation might be, it appears that you’re meant to calculate the three matrix product and then compute their determinants instead of computing $A$ and $B$ directly, which would be much less work.