**4**

The secret life of determinants

**4.1 The area of a parallelogram**

Let us investigate the behaviour of the area (**a**, **b**) of the parallelogram with vertices **0**, **a**, **a** + **b** and **b**.

Our first observation is that it appears that

To see this, examine Figure 4.1 showing the parallelogram *OAXB* having vertices *O* at **0**, *A* at **a**, *X* at **a** + **b** and *B* at **b** together with the parallelogram *OAX′B′* having vertices *O* at **0**, *A* at **a**, *X′* at (1 + *λ*)**a** + **b** and *B′* at *λ***a** + **b**. Looking at the diagram, we see that (using congruent triangles)

and so

A similar argument shows that

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