Find the determinant of the matrix A with this linear transformation.

Question

Consider a linear transformation $T(x) = Ax$ from $\Bbb R^2$ to $\Bbb R^2$. Suppose for two vectors $\vec v_1$ and $\vec v_2$ in $\Bbb R^2$ we have $T(\vec v_1) = 5 \vec v_2$ and $T(\vec v_2)=-6 \vec v_1$. Find the determinant of the matrix $A$.

Answer 1

Clearly, you need to assume $v_1,v_2$ are nonzero. Then $v_1$ and $v_2$ are independent, because otherwise $v_1=kv_2$ and then $-6k^2 v_2 =k Av_2 = Av_1= 5v_2 $ which gives $k^2<0$.

Then write $A$ in the basis $(v_1,v_2)$, and it has the form $$\pmatrix{0 & 5 \\ -6 & 0}$$ so the determinant is $30$.