What is a matrix transformation?

A matrix transformation maps every point (x,y) to a new point using a 2x2 matrix. It can represent rotation, scaling, shearing, and reflection — all fundamental operations in linear algebra and computer graphics.

What does the determinant of a 2x2 matrix tell you?

The determinant (ad - bc) tells you how the matrix scales area. |det| is the area scale factor. A positive determinant preserves orientation, negative flips it (mirror), and zero means the transformation collapses space to a line or point.

How do basis vectors work in matrix transformations?

The columns of a 2x2 matrix are where the basis vectors e₁=(1,0) and e₂=(0,1) land after the transformation. This means you can read a matrix's effect directly from its columns.

Matrix Transformer

Enter a 2×2 matrix or pick a preset. Watch the unit square transform with animated basis vectors and a warped grid.

Matrix

A =

Presets

Properties

det(A)	1.0000
	Preserves orientation

The Math Behind It

Linear Transformation

A 2×2 matrix maps every point (x,y) to a new point (x',y')
x' = ax + by, y' = cx + dy
Columns of the matrix are the images of the basis vectors e₁ and e₂
The red arrow shows where (1,0) lands; blue shows where (0,1) lands

Determinant

det(A) = ad - bc
|det| = area scale factor of the transformation
det > 0: orientation preserved
det < 0: orientation flipped (mirror)
det = 0: transformation collapses to a line or point

Matrix Transformer

Matrix

Properties

The Math Behind It

Linear Transformation

Determinant

Related Visualizations

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Matrix Calculator

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