To solve this problem, we use the property of determinants related to row or column scaling. Specifically, if we multiply one row or column of a matrix by a scalar, the determinant of the matrix is also multiplied by that scalar.

Given:

• $\det(A) = 7$, where $A$ is a $3 \times 3$ matrix.
• Matrix $B$ is obtained from $A$ by multiplying the second column by 2.

Determinant property:

If we multiply any column (or row) of a matrix by a scalar $k$, then the determinant of the matrix is multiplied by $k$. Here, since we multiplied the second column by 2, the determinant of $B$ will be twice the determinant of $A$.

Calculation:

$\det(B) = 2 \times \det(A) = 2 \times 7 = 14$

Thus, $\det(B) = 14$.

Let me know if you'd like further details or have any questions!

Here are 5 related questions for further exploration:

1. What happens to the determinant if both rows and columns are scaled by the same factor?
2. How does swapping two rows or columns of a matrix affect its determinant?
3. What is the determinant of a diagonal matrix?
4. How does the determinant change if a row or column is added to another row or column?
5. Can we infer anything about the determinant if one row of a matrix is all zero?

Tip: The determinant of a matrix is zero if two rows or columns are identical, or if one row or column is all zeros.