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]