Question

Determine whether the linear transformation is invertible. If it is, find its inverse. (If an answer does not exist, enter DNE.) T(x, y) = (-x + y, -x - y)

Answer 1

Answer:

$((-\frac{1}{2}(x+y), -\frac{1}{2} (x-y) )$

Step-by-step explanation:

T(x , y) = (-x -y, -x +y)

T(1, 0) = (-1, -1) = -1(1 , 0) -1(0 , 1)

T(0, 1) = (-1, 1) = -1(1 , 0) +1(0 , 1)

Therefore,

$T = \left[\begin{array}{ccc}-1&-1\\-1&1\end{array}\right]$

|T| = [-1 -1] = -2

T is invertible

$T^-^1 = -\frac{1}{2} \left[\begin{array}{ccc}1&+1\\+1&-1\end{array}\right] \\\\=\left[\begin{array}{ccc}-1/2&-1/2\\-1/2&1/2\end{array}\right]$

Therefore,

$T^-^1(x,y)=(-\frac{1}{2}x -\frac{1}{2} y,-\frac{1}{2} x+\frac{1}{2} y)$

$((-\frac{1}{2}(x+y), -\frac{1}{2} (x-y) )$