​$$\text{If } A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 7 \end{bmatrix} \text{ and } B = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{bmatrix}, \text{ then } \det(A + B) = ?$$​​

Correct option is A

​$$\text{First, calculate } A + B: \\A + B = \begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix} \\\text{Now, compute } \det(A + B): \\\det\begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix} = 2 \left( \det \begin{bmatrix} 6 & 7 \\ 9 & 10 \end{bmatrix} \right) - 3 \left( \det \begin{bmatrix} 5 & 7 \\ 8 & 10 \end{bmatrix} \right) + 4 \left( \det \begin{bmatrix} 5 & 6 \\ 8 & 9 \end{bmatrix} \right) \\\text{Calculate the 2x2 determinants:} \\\det \begin{bmatrix} 6 & 7 \\ 9 & 10 \end{bmatrix} = (6)(10) - (7)(9) = 60 - 63 = -3 \\\det \begin{bmatrix} 5 & 7 \\ 8 & 10 \end{bmatrix} = (5)(10) - (7)(8) = 50 - 56 = -6 \\\det \begin{bmatrix} 5 & 6 \\ 8 & 9 \end{bmatrix} = (5)(9) - (6)(8) = 45 - 48 = -3 \\\text{Substitute into the determinant expression:} \\\det(A + B) = 2(-3) - 3(-6) + 4(-3) \\\det(A + B) = -6 + 18 - 12 = 0$$​​