Question

Consider the real matrices

Hint:

Use determinant properties such as scaling of rows/columns and transpose invariance: \[ \det(A^T)=\det(A). \]

Answer 1

Correct Answer: 4

Step 1: Link $B$ to $C$.
The rows of $B$ are the first row of $C$, the vector $(1,-3,5)$, and twice the second row of $C$.

Step 2: Pull out the scalar.
The factor $2$ in the third row gives $\det B=2\det\begin{vmatrix}a&b&c\\1&-3&5\\d&e&f\end{vmatrix}$.

Step 3: Swap rows to align with $C$.
Swapping rows $2$ and $3$ introduces a sign and, after matching the third row with that of $C$, gives the relation $\det C=2\det B$.

Step 4: Plug in.
With $\det B=2$, $\det C=2\cdot2=4$.

Step 5: Conclude.
\[ \boxed{4} \]