Suppose $3\times3$ matrix $A = \left[\begin{smallmatrix} 1 & 2 & 2 \\ 0 & 1 & 1 \\ 0 & 1 & 2 \end{smallmatrix}\right]$. And suppose $T: \mathbb R^3 \to \mathbb R^3$ be defined by $T(x) = A(x)$ for every $x$ in $\mathbb R^3$.
- Find $A^{-1}$ (Easy)
- Suppose $T^{-1}$ be the inverse transformation $T$, $y = T^{-1}$ and suppose $x = (1,1,1)$. Find this inverse transformation and use it to evaluate $T^{-1}(x)$
- Suppose $y = T^{-1}(x)$. Evaluate $T(y)$.
Can anyone help me understand 2. and 3.? Is this a trick question since $T(x) = A(x)$?
For (b) you need to know that $T^{-1}(\vec{x}) = A^{-1}\vec{x}$. So you can get $T^{-1}(\vec{x})$ by multiplying the inverse from part (a) with your vector $(1,1,1)^t$.
Yes, (c) is a trick question: remember that the composition of a function and its inverse is always the identity function. $$ T(\vec{y})=T(T^{-1}(\vec{x})) = A(A^{-1}\vec{x}) = (AA^{-1})\vec{x} = I\vec{x} = \vec{x} $$