Problem on matrices as transformations - Leading Lesson

Problem on matrices as transformations

Let $A =\begin{pmatrix}1 & 2 \\ 2 & 1\end{pmatrix}$.

How is the rectangle between $(0,0), (2,0), (2,1),$ and $(0,1)$ transformed by application of $A$?

Solution
Recall that
Matrices as transformations

One way to determine how a shape is transformed by application of a matrix is to directly compute the matrix multiplication on several key points of that shape.
We see how the rectangle is transformed by computing the matrix multiplications $$A \begin{pmatrix}0 \\ 0 \end{pmatrix}, A \begin{pmatrix}2 \\ 0 \end{pmatrix}, A \begin{pmatrix}2 \\ 1 \end{pmatrix}, \text{ and } A \begin{pmatrix}0 \\ 1 \end{pmatrix}$$
Matrix multiplication

Computing, we get \begin{align}
A \begin{pmatrix}0 \\ 0 \end{pmatrix} &= \begin{pmatrix}0\\0\end{pmatrix}\\
A \begin{pmatrix}2 \\ 0 \end{pmatrix} &= \begin{pmatrix}2\\4\end{pmatrix}\\
A \begin{pmatrix}2 \\ 1 \end{pmatrix} &= \begin{pmatrix}4\\5\end{pmatrix}\\
A \begin{pmatrix}0 \\ 1 \end{pmatrix} &= \begin{pmatrix}2\\1\end{pmatrix}
\end{align}
We see that the four corners of the rectangle are transformed to $(0,0), (2,4), (4,5)$, and $(2,1)$.
We conclude that the rectangle gets mapped to the following parallelogram:

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