The matrix \(\mathbf { S } = \left( \begin{array} { l l } - 1 & 2 - 3 & 4 \end{array} \right)\) represents a transformation.
(A) Show that the point \(( 1,1 )\) is invariant under this transformation.
(B) Calculate \(\mathbf { S } ^ { - 1 }\).
(C) Verify that \(( 1,1 )\) is also invariant under the transformation represented by \(\mathbf { S } ^ { - 1 }\).
Part (i) may be generalised as follows.
If \(( x , y )\) is an invariant point under a transformation represented by the non-singular matrix \(\mathbf { T }\), it is also invariant under the transformation represented by \(\mathbf { T } ^ { - 1 }\).
Starting with \(\mathbf { T } \binom { x } { y } = \binom { x } { y }\), or otherwise, prove this result.