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Determinants — Part 1: Evaluating determinants, areas and collinearity

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Lesson

Determinants, Part 1: one number for a square matrix

Coordinates in, area out: a determinant turns a square array into a single number that measures size.

In this lesson you will

Evaluate determinants of order 1, 2 and 3
Expand a 3 × 3 determinant along any row or column
Solve equations involving determinants
Find the area of a triangle from its vertices
Test collinearity and find the equation of a line using determinants

$PQ$
$QR$
$PR$

Triangle $PQR$: area $= \frac12|16| = 8$ square units.

Concept

A determinant is a number

For a square matrix $A$ , its determinant $|A|$ (or $\det A$ ) is a single number. Only square matrices have one.

Every square matrix has exactly one number attached to it: its determinant.
Square matrix	Its determinant	Value
$\begin{bmatrix} 7 \end{bmatrix}$	$\|7\|$	$7$
$\begin{bmatrix} 5 & -3 \\ 4 & 2 \end{bmatrix}$	$\begin{vmatrix} 5 & -3 \\ 4 & 2 \end{vmatrix}$	$22$
$\begin{bmatrix} 2 & -1 & 3 \\ 1 & 4 & 0 \\ -2 & 1 & 5 \end{bmatrix}$	$\begin{vmatrix} 2 & -1 & 3 \\ 1 & 4 & 0 \\ -2 & 1 & 5 \end{vmatrix}$	$72$