Determinants

The determinant of a square matrix is a single number that measures how much the associated linear map scales volumes. For a $2\times 2$ matrix, $\det \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=ad-bc$. For larger matrices, expand along any row or column: multiply each entry by its cofactor and add the results. A matrix with a row or column of zeros has determinant zero, and triangular matrices have determinant equal to the product of their diagonal entries. If $A$ is $n\times n$ and $c$ is a scalar, then $\det (cA)=c^n\det A$. Applying a map twice scales volume by $(\det T{)}^2$.