Discussion
Let \(T : \mathbb{R}^{2} \to \mathbb{R}^{2}\) be a linear transformation defined by
T((1, 2)) = (1, 0) and T((2, 1)) = (1, 1).
For \(p, q \in \mathbb{R}\), let \(T^{-1}((p, q)) = (x, y)\).
Which of the following statements is TRUE?
Answer: B
The inverse is x = p + q; y = 2p - q.
Read the given data backwards: \(T^{-1}(1,0) = (1,2)\) and \(T^{-1}(1,1) = (2,1)\). Subtracting, \(T^{-1}(0,1) = (2,1) - (1,2) = (1,-1)\). By linearity
\[ \begin{aligned} T^{-1}(p,q) &= p(1,2) + q(1,-1) \\ &= (p + q,\ 2p - q) \end{aligned} \]
Check with (p, q) = (1, 1): x = 2, y = 1, which is (2, 1) as required. The option x = p - q fails this check, since it gives x = 0, and y = 2p + q gives y = 3, so the other three options are wrong.
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