# 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.