# Determinants

The determinant det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix. Determinants are named after the size of the matrices. In the following example we will show how to determine the second order determinants.

Example

$A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}$

The determinant of A(second order determinant) is

$det(A)=\begin{vmatrix} a & b\\ c & d \end{vmatrix}=ad-bc$

Determinants of 3 × 3 matrices are called third-order determinants. One method of evaluating third-order determinants is called expansion by minors.  The minor of an element is the determinant formed when the row and column containing that element are deleted.

$\begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}=a\begin{vmatrix} e & f\\ h & i \end{vmatrix}-b\begin{vmatrix} d & f\\ g & i \end{vmatrix}+c\begin{vmatrix} d & e\\ g & h \end{vmatrix}$

Determinants could be used to find the area of a triangle if the coordinates of the vertices are known. If a triangles vertices are (a,b), (c,d) and (e,f) the area is

$A=\frac{1}{2}\begin{vmatrix} a & b & 1\\ c & d & 1\\ e & f & 1 \end{vmatrix}$

If A  turns out to be negative then we must use the absolute value for A to have a nonnegative value for our area.

Example

Find the area of an triangle with its vertices located at (-2,2), (1,3) and (3,0) (this example is also shown in our video lesson).

We plug our coordinates for the vertices into our area formula

$A=\frac{1}{2}\begin{vmatrix} -2 & 2 & 1\\ 1 & 3 & 1\\ 3 & 0 & 1 \end{vmatrix}$

and continues with

$=\frac{1}{2}(-2\begin{vmatrix} 3 & 1\\ 0 & 1 \end{vmatrix}-2\begin{vmatrix} 1 & 1\\ 3 & 1 \end{vmatrix}+1\begin{vmatrix} 1 & 3\\ 3 & 0 \end{vmatrix})=\\ \\ =\frac{1}{2}(-2(3\cdot1-1\cdot 0 )-2(1\cdot 1-1\cdot 3)+1(1\cdot 0-3\cdot 3))=\\ \\ =\frac{1}{2}(-6+4-9)=\frac{-11}{2}=-5.5$

We received a negative value for A and an area cannot be negative, therefore we must take the absolute value for A:

$\mid A\mid =\mid -5.5\mid =5.5\;$

So the triangles area is 5.5 square units.

## Video lesson

Find the area from the example above.