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3. Determinants

Important: $ \det A$ makes sense only if $ A$ is square! The corresponding linear transformation $ T _ A$ therefore is from $ n$-space of some dimension $ n$ to itself:
$ T _ A:$   R$ ^ n \rightarrow$   R$ ^n$.



Interpretations of the determinant (assuming $ \det A \neq 0$):

If $ A$ is $ 2 \times 2$, $ \vert\det A\vert$ is the factor by which the corresponding linear transformation changes all areas.

If $ A$ is $ 3 \times 3$, $ \vert\det A\vert$ is the factor by which the corresponding linear transformation changes all volumes.

In each case, $ \det A > 0$ if and only if the transformation preserves orientation. In the $ 2 \times 2$ case, this means that figures in the domain are not flipped over; in the $ 3 \times 3$ case, this means that a right-hand glove is not turned into a left-hand glove.

Correspondingly, $ \det A < 0$ if and only if the transformation reverses orientation. Finally, $ \det A = 0$ if and only if the transformation collapses all areas or volumes to zero, in which case $ A$ is singular.



Review the mechanics of determinants on your own--both calculations using cofactors and calculations using row-reduction. Unless the matrix you are interested in is very small or has many zero entries, by hand it is fastest to use row-reduction. Except in the $ 2 \times 2$ case, the full expansion with permutations is not usually the best method.



next up previous
Next: e_htls Up: e_htls Previous: e_htls
Kirby A. Baker 2002-01-10