Editing Determinant (section)

=== First properties ===
The determinant has several key properties that can be proved by direct evaluation of the definition for <math>2 \times 2</math>-matrices, and that continue to hold for determinants of larger matrices. They are as follows:<ref>{{harvnb|Lang|1985|loc=§VII.1}}</ref> first, the determinant of the [[identity matrix]] <math>\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}</math> is 1.
Second, the determinant is zero if two rows are the same:
:<math>\begin{vmatrix} a & b \\ a & b \end{vmatrix} = ab - ba = 0.</math>
This holds similarly if the two columns are the same. Moreover,
:<math>\begin{vmatrix}a & b + b' \\ c & d + d' \end{vmatrix} = a(d+d')-(b+b')c = \begin{vmatrix}a & b\\ c & d \end{vmatrix} + \begin{vmatrix}a & b' \\ c & d' \end{vmatrix}.</math>
Finally, if any column is multiplied by some number <math>r</math> (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:
:<math>\begin{vmatrix} r \cdot a & b \\ r \cdot c & d \end{vmatrix} = rad - brc = r(ad-bc) = r \cdot \begin{vmatrix} a & b \\c & d \end{vmatrix}.</math>