Injective Linear transformation .

Question

If the linear transformation $T:\mathbb R^3\to\mathbb R^3$ defined as $$ T(x,y,z)=(y+kz,x+ky,x-2y+z) $$ is injective, what is the value of $k$?

Answer 1

Notice how T is represented by the matrix \begin{bmatrix}0&1&k\\1&k&0\\1&-2&1\end{bmatrix}

Now calculating the determinant with the formula of Sarrus:

$$-2 k-k²-1=\det(T) = -(k+1)²$$

If the determinant is zero, then $T$ can't be injective. If the determinant is nonzero, then $T$ is injective.

$k$ can't be $-1$.