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

`((-(1)/(2)(x+y), -(1)/(2) (x-y) )`

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 = -(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)=(-(1)/(2)x -(1)/(2) y,-(1)/(2) x+(1)/(2) y)`

`((-(1)/(2)(x+y), -(1)/(2) (x-y) )`