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Think of the matrix $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ as mapping one plane to another.
a) If two lines in the first plane are parallel, show that after being mapped by $A$ they are also parallel - although they might coincide.
b) Let $Q$ be the unit square: $0<x<1,0<y<1$ and let $Q^{\prime}$ be its image under this map A. Show that the area $\left(Q^{\prime}\right)=|a d-b c|$. [More generally, the area of any region is magnified by $|a d-b c|$ ( $a d-b c$ is called the determinant of a $2 \times 2$ matrix]
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