The diagram illustrate a scaling in the x – direction. If the x -values increase by a factor of 3, then we can represent the transformation by the matrix

If the corners of the rectangle are atandthen the new vertices will be at

respectively.

The scale factor in the x – direction is 3 and there is no scaling in the y – direction so the new area is 3 times the old one.

A scaling in the y direction is illustrated below.

If the y – values increase by a factor of 5, then we can represent the transformation by the matrix

If the corners of the rectangle are atandthen the new vertices will be at

respectively.

The scale factor in the y – direction is 5 and there is no scaling in the x – direction so the new area is 5 times the old one.

We can combine scalings in the x – direction with scalings in the y direction, and represent them both in a single matrix. A scaling in the x direction by 3 combined with a scaling in the y direction by 5 can be represented by the matrixThis is illustrated below.

If the corners of the rectangle are atandthen the new vertices will be at

respectively.

Ths scale factor in the x – direction is 3 and the scale factor in the y – direction is 5 so the new area is 3*5=15 times the area of the old one.