Solution

Given:

$T(x,y,z) = (2x,\; x-y,\; 5x+4y+z)$

Write system:

$u = 2x$

$v = x - y$

$w = 5x + 4y + z$

From first:

$x = \frac{u}{2}$

From second:

$y = x - v = \frac{u}{2} - v$

From third:

$z = w - 5x - 4y$

$z = w - \frac{5u}{2} - 4\left(\frac{u}{2} - v\right)$

$z = w - \frac{5u}{2} - 2u + 4v$

$z = w - \frac{9u}{2} + 4v$

Thus inverse:

$T^{-1}(x,y,z) = \left(\frac{x}{2}, \frac{x - 2y}{2}, \frac{-9x + 8y + 2z}{2}\right)$