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# Determinant

<figure><img src="https://1983523492-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F0nwEx8a60yETwfNYnenT%2Fuploads%2FSdWYxduDLl4ZrlkwCi32%2Fimage.png?alt=media&amp;token=ba0bc29e-52ce-4709-ab7a-e469bbb80801" alt=""><figcaption></figcaption></figure>

### Properties of the Determinant

For example, the determinant of a transformation would be $$33$$ if that transformation increases the area of a region by a factor of 3.

<figure><img src="https://1983523492-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F0nwEx8a60yETwfNYnenT%2Fuploads%2FWgmSfnPR6iZGOLbYRU9M%2Fimage.png?alt=media&amp;token=85dde6ee-1ec8-430c-ad87-0786627e6fff" alt=""><figcaption></figcaption></figure>

The determinant of a 2D transformation is $$0$$ if it squishes all of space onto a line, or even onto a single point, since the area of every region would then become 0.

That last one is especially important; checking if the determinant of a given matrix is $$00$$ will give a way of computing whether or not the transformation associated with a matrix squishes everything into a smaller dimension.&#x20;

{% hint style="info" %}
Determinant for 3D matrics affect volume instead of area
{% endhint %}