Question

Let \( M\in M_3(\mathbb{R}) \). If

Hint:

If \(MA=B\), then determinants satisfy \(\det(M)\det(A)=\det(B)\). This is the fastest way to compute determinants in transformation problems.

Answer 1

Correct Answer: -0.5

Step 1: Stack the vectors into matrices.
Put the three input vectors as columns of $A$ and their images as columns of $B$. Then the map $M$ satisfies $MA=B$.

Step 2: Take determinants.
\[ \det(M)=\frac{\det(B)}{\det(A)} \]

Step 3: Determinant of $A$.
Pulling out the common roots from the columns and computing the leftover $3\times3$ determinant gives $\det(A)=-30\sqrt{30}$.

Step 4: Determinant of $B$.
Doing the same factoring and simplification for $B$ gives $\det(B)=15\sqrt{30}$.

Step 5: Divide.
\[ \det(M)=\frac{15\sqrt{30}}{-30\sqrt{30}}=-\frac12 \]
\[ \boxed{-0.5} \]