Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online course: Linear Algebra - Matrices - Determinants

2-by-2 Determinant

[previous page] [next page] [table of contents][page overview]
The determinant of a $ 2\times2$-matrix is

$\displaystyle \left\vert\begin{array}{cc} a & b \\ c & d \end{array}\right\vert = ad - bc. $

This can easily be seen from the explicit formula for determinants as a sum over permutations, since in this case there are only $ 2$ addends.

Alternatively, the determinant can be computed by means of its defining properties. Representing the columns as linear combinations of the canonical unit vectors it follows from multilinearity:

$\displaystyle \operatorname{det}(a e_1 + c e_2, b e_1 + d e_2)$ $\displaystyle =$ $\displaystyle a \operatorname{det}(e_1,be_1+de_2)+ c \operatorname{det}(e_2,be_1+de_2)$  
  $\displaystyle =$ $\displaystyle ab \operatorname{det}(e_1,e_1)+ ad \operatorname{det}(e_1,e_2)$  
    $\displaystyle +\, cb \operatorname{det}(e_2,e_1)+ cd \operatorname{det}(e_2,e_2)$  
  $\displaystyle =$ $\displaystyle ab\cdot 0+ad\cdot 1+cb\cdot(-1)+cd\cdot0 = ad-bc$  

  automatically generated 4/21/2005