Matrix Determinant

Table of Contents

</aside>

Determinant inputs a square matrix$(n {\times} n)$ and outputs a real number.

What’s a Determinant?

Determinant is a single number calculated from a square matrix.

This single number tells us important properties about the transformation represented by the matrix </aside>

Formula

Here is $2 {\times} 2$ square matrix.

$$ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$
The determinant of $A$ matrix.

$$ det(A) = |A| = ad - bc $$

Example

matrix $A$

$$ A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} $$
The determinant of matrix $A$.

$$ det(A) = |A| = (2 \times 4) - (3 \times 1) = 8 - 3 = 5 $$

The most intuitive understanding of determinants comes from their geometric interpretation.

Change in Area/Volume (Δ)

Determinant of $2 {\times} 2$ matrix represents how much the area of a unit square changes after transformation.
For $3 {\times} 3$ matrix, it shows the change in volume of a unit cube.

Change in Orientation

Positive

Negative

If the determinant is negative, the transformation flips the orientation of space.

데스모스 링크

Linear Transformations

</aside>

Orientation Preservation