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

The determinant equals the signed area of the parallelogram formed by the columns of the matrix. A positive determinant means the transformation preserves orientation; a negative determinant flips it. A zero determinant means the matrix is singular and collapses the plane to a line or a point.

How do you classify a 2x2 linear transformation?

Common types include rotations (preserve length and angles, determinant 1), reflections (determinant negative 1, swap orientation), shears (determinant 1 but distort angles), scalings (multiply lengths by a constant), and singular maps that collapse dimensions. The determinant, trace, and eigenvalues together determine the class.

What happens when the determinant equals zero?

When the determinant is zero the matrix is singular: it cannot be inverted and it collapses the plane onto a line through the origin (rank 1) or onto the origin itself (rank 0, only when A is the zero matrix). The kernel and image become visible structural features in this case.

How are eigenvalues related to linear transformations?

An eigenvalue lambda paired with an eigenvector v satisfies A times v equals lambda times v. Geometrically, the eigenvector direction is preserved by the transformation and only scaled by lambda. Real eigenvalues produce visible invariant lines; complex eigenvalues indicate a rotational component with no real invariant direction.

Linear Transformation

Linear Algebra·Linear transformationsA 2×2 matrix reshapes the plane — pick a scenario, scrub t from 0 to 1, and read what is happening on the right.

det M(t)=1

classify=identity

▸Animationt = 0.000

t = 0.000

Identity

rank 2 · det 1

The identity matrix maps every vector to itself: Iv = v. The grid stays exactly where it started.

Insight

Both columns are the standard basis vectors. Every direction is an eigenvector with eigenvalue 1.

What to watch

Edit a single entry to break the identity — even a small perturbation gives a non-trivial map.
Learn more about the identity · all twelve presets

AMatrix

04Livet = 0.000

M(t)

1001

det M(t)1det A1trace A2eigenvalues1, 1rank A2

Key Terms

Getting Started

The Animation Slider

Preset Scenarios — Three Rank Categories

The Identity: the Transformation That Does Nothing

Full Rank: Reshaping Without Losing Anything

Rank 1: Collapsing the Plane Onto a Line

Rank 0: the Zero Map

Editing the Matrix A Directly

Display Layer Toggles

Reading the Live Card

The Classification Readout

What Is a Linear Transformation?

Determinant: Area and Orientation

Eigenvalues, Rank, and Matrix Types

Related Concepts

Key Terms

Linear Transformation — A function

T

such that

T(u + v) = T(u) + T(v)

and

T(c \cdot v) = c \cdot T(v)

. Every linear transformation on

\mathbb{R}^2

is represented by a

2 \times 2

matrix.

Matrix $A$ — The

2 \times 2

array whose columns are

T(\hat{i})

and

T(\hat{j})

, where

\hat{i}

and

\hat{j}

are the standard basis vectors.

Determinant — The signed area of the parallelogram spanned by the columns of

A

. Positive: orientation preserved. Negative: orientation reversed. Zero: singular.

Trace — The sum of the diagonal entries,

a_{11} + a_{22}

. Equals the sum of the eigenvalues.

Rank — The dimension of the image. Either 2 (full rank, invertible), 1 (singular, collapses to a line), or 0 (zero matrix).

Eigenvalue — A scalar

\lambda

such that

Av = \lambda v

for some nonzero

v

. The directions preserved by

A

up to scaling.

Interpolation Parameter $t$ — A scrub variable from 0 to 1. The displayed matrix is

M(t) = (1 - t) I + t A

, morphing the identity into

A

Getting Started

The visualizer shows a single square canvas with three columns surrounding it. The left sidebar groups twelve preset transformations by rank. The center column holds the canvas and animation controls. The right column shows an explanation of the current preset, layer toggles, the editable matrix

A

, and a live readout.

The fundamental workflow:

• Pick a preset on the left — the animation auto-plays, morphing the identity into the chosen matrix over about 1.6 seconds.
• Drag the t slider under the canvas — manually scrub through the morph at any speed.
• Edit the matrix A on the right — the visualization updates instantly. The scenario indicator clears since you are now in custom mode.

The matrix

M(t) = (1 - t) I + t A

blends linearly from the identity at

t = 0

to the full transformation at

t = 1

, so the canvas shows the transformation actually happening rather than just its end state.

The Animation Slider

The animation panel below the canvas controls the parameter

t

from 0 to 1.

• Play — smoothly eases from current

t

to 1 over 1.6 seconds using a cubic ease-in-out curve. Restarts from 0 if already at the end.
• Step buttons — advance or rewind by 0.1 in

t

for frame-by-frame inspection.
• Reset — jump back to

t = 0

(identity).
• Scrub slider — drag to any value between 0 and 1. Hold and slow-scrub to watch the determinant cross zero in singular cases.

Watch the readout under the canvas as you scrub:

\det M(t)

updates live. For singular target matrices,

\det M(t)

smoothly approaches zero. For orientation-reversing matrices (reflections),

\det M(t)

crosses zero somewhere between 0 and 1, marking the instant the plane collapses before flipping.

Preset Scenarios — Three Rank Categories

The scenarios sidebar offers twelve canonical transformations grouped by rank.

• Full rank (rank 2) — identity, rotate

45\deg

, rotate

90\deg

, scale

2\times

, shear x, reflect x-axis, reflect y = x, and twist & stretch. All invertible. Different determinants and orientation behaviors.
• Singular (rank 1) — project to x-axis, project to y = x (orthogonal projection), and outer product

[[1,2],[2,4]]

. Each collapses the plane to a line; kernel and image become visible.
• Degenerate (rank 0) — the zero map. Everything goes to the origin.

Selecting a scenario auto-plays the morph from identity to that matrix. The explanation card on the right shows three blocks: a brief description, an Insight about structural properties (eigenvalues, symmetries), and What to watch — the specific visual feature that makes this scenario distinctive.

The Identity: the Transformation That Does Nothing

The Identity preset sets

A = I

, and the canvas shows a plane that has not moved. The transformed grid sits exactly on the original,

hat{i}

still points to

(1, 0)

and

hat{j}

still points to

(0, 1)

.

It is worth starting here because it establishes what every later picture is a deviation *from*. Whatever a preset does, it does it to this.

Identity, frozen at t = 1

Nothing has moved: the transformed grid lies on the original, and the basis arrows still point to (1, 0) and (0, 1). Every preset shows this same picture at t = 0.

The identity is also the left end of the scrub slider. The displayed matrix is

M(t) = (1-t)I + tA

, so at

t = 0

every preset shows this same picture, and the animation is a straight-line morph from here to

A

.

Two properties fall out immediately. Every non-zero vector is an eigenvector with eigenvalue 1, since

Iv = 1 cdot v

holds for all

v

. And

det I = 1

, so areas and orientation are untouched — which is the baseline the determinant readout measures against.

Full Rank: Reshaping Without Losing Anything

Eight of the twelve presets are full-rank maps — rotations, scalings, shears, reflections and the general twist. Their common property is $det A
eq 0$, which means the transformed grid stays a genuine grid: two independent directions in, two independent directions out.

The frozen picture below shows the 45° rotation at

t = 1

. The unit circle is still a circle, the unit square has become a tilted square of equal area, and no direction has been flattened.

Rotate 45°, frozen at t = 1

A full-rank map. The unit circle is still a circle and the unit square is a tilted square of equal area - det = 1, so nothing is stretched or flattened.

Within this group the determinant separates the sub-cases. Rotations and shears have

det = 1

and preserve area exactly; the

2 imes

scaling has

det = 4

and quadruples it; the reflections have

det = -1

, preserving area but reversing orientation — watch

hat{i}

and

hat{j}

swap handedness.

The essential point is invertibility. A full-rank map can be undone, because no two distinct vectors are sent to the same place. The kernel contains only the origin, and the image is the entire plane — which is why the kernel and image layers have nothing interesting to draw on any of these eight presets.

Rank 1: Collapsing the Plane Onto a Line

Three presets are singular with rank 1: projection onto the x-axis, projection onto the line

y = x

, and the outer-product matrix

egin{pmatrix} 1 & 2 \ 2 & 4 end{pmatrix}

, whose second column is twice its first.

Here

det A = 0

and the whole plane is flattened onto a single line through the origin. The frozen picture shows the projection onto the x-axis: the transformed grid has become one line, and the unit circle has become a segment.

Project to the x-axis, frozen at t = 1

Rank 1: the whole plane has collapsed onto one line. The green image line is where everything lands; the red kernel line is everything sent to the origin.

This is where the kernel and image layers earn their place. The image is the line everything lands on — drawn in green. The kernel is the line of vectors sent to the origin — drawn in red. Both are one-dimensional, and the rank-nullity theorem says their dimensions must sum to 2, which is exactly what the picture shows.

Information is destroyed here, irreversibly. Every point on a given kernel-parallel line collapses to the same image point, so there is no way back and no inverse exists. That is what a zero determinant means in practice, and it is why "singular" and "non-invertible" are the same statement.

Rank 0: the Zero Map

One preset takes the collapse to its limit. The zero matrix sends every vector to the origin, so the transformed grid, the unit square and the unit circle all contract to a single point.

The frozen picture is that end state: nothing remains but the origin dot.

The zero map, frozen at t = 1

Rank 0. Grid, square and circle have all contracted to the origin - the image is a single point and the kernel is the entire plane.

The rank-nullity accounting still holds, at the other extreme. The image is the origin alone — dimension 0 — and the kernel is the entire plane — dimension 2. They sum to 2, as they must.

This preset is the reason "rank" is a more informative measure than "invertible or not". The projections and the zero map are all singular, but they are not equally destructive: a rank-1 map keeps one dimension of information and a rank-0 map keeps none. The determinant reports

0

for both and cannot tell them apart.

Editing the Matrix A Directly

The matrix card on the right shows the four entries of

A

as editable numeric inputs.

• Click an entry and type — the matrix updates instantly. Use arrow keys to step by 0.1.
• Set both columns parallel —

\det A = 0

, the matrix becomes singular, kernel and image lines appear on the canvas.
• Set all four entries to zero — the zero map. The entire plane collapses to the origin.
• Make the matrix orthonormal — for example

[[\cos\theta, -\sin\theta], [\sin\theta, \cos\theta]]

— the classification readout shows "rotation".

When you edit, the active scenario indicator clears and the explanation card switches to "Custom matrix". Click any scenario in the sidebar to snap back to a canonical example.

Display Layer Toggles

The chip strip toggles eight independent visual layers.

• grid — the transformed integer grid. The deformation of the gridlines is the most direct picture of what

M(t)

does.
• basis — the two basis vectors

\hat{i}

(orange) and

\hat{j}

(cyan) drawn from origin to their images. The columns of

M(t)

are these tips.
• unit sq — the unit square fill. Purple when

\det > 0

, pink when

\det < 0

(orientation reversed), gray when singular.
• unit circle — the unit circle morphs into an ellipse for invertible

A

. Ellipse axes correspond to singular values.
• eigen — magenta dashed lines along eigendirections (when eigenvalues are real). Tips show how eigenvectors scale.
• ker / im — red dashed kernel line and green image line, shown only for singular matrices.
• samples — a scattered grid of dots showing where representative points land.
• labels — the

\hat{i}

\hat{j}

\ker A

\text{im } A

tags.

Reading the Live Card

The live card on the right reports six quantities updating in real time.

• $M(t)$ — the current interpolated matrix shown as a

2 \times 2

array. At

t = 0

this is the identity; at

t = 1

this is

A

.
• $\det M(t)$ — the determinant of the currently displayed matrix. Watch it deform from 1 (identity) toward

\det A

.
• $\det A$ — the determinant of the target matrix. Fixed.
• trace $A$ — sum of diagonal entries. Equals the sum of eigenvalues.
• eigenvalues — the roots of the characteristic polynomial. Listed as real pairs or as

a \pm bi

for complex conjugates.
• rank $A$ — the dimension of the image: 0, 1, or 2.

The combination of

\det A

, trace, and eigenvalues uniquely characterizes the structural type of the transformation up to similarity.

The Classification Readout

Below the canvas the classify panel names the transformation in plain language, matched against ten patterns.

• identity —

A = I

. Nothing moves.
• uniform scaling — diagonal with equal positive entries.
• axis-aligned stretch — diagonal with unequal entries. Scales each axis independently.
• rotation — columns are unit length and perpendicular,

\det = 1

.
• reflection — columns are unit length and perpendicular,

\det = -1

.
• orientation-reversing —

\det < 0

but not a pure reflection.
• general invertible —

\det \neq 0

, no special structure.
• singular → line — rank 1, collapses the plane to a line.
• zero map — everything goes to the origin.

Edit the matrix and watch the classification update. Tiny perturbations break special structure: a rotation becomes "general invertible" when one entry shifts by 0.01.

What Is a Linear Transformation?

A linear transformation is a function

T: V \to W

between vector spaces satisfying two rules:

T(u + v) = T(u) + T(v)

T(c \cdot v) = c \cdot T(v)

These two properties together say

T

respects the linear combinations of vectors. Equivalently,

T

maps lines through the origin to lines through the origin, and grids of parallelograms to grids of parallelograms.

For finite-dimensional spaces, every linear transformation is given by matrix multiplication. In

\mathbb{R}^2

, the matrix has the images

T(\hat{i})

and

T(\hat{j})

as its columns. That is the entire content of "linear": fix where the basis vectors go, and everything else is determined.

For deeper coverage of definitions and properties, see linear transformation theory, linear maps, and vector spaces.

Determinant: Area and Orientation

The determinant of a

2 \times 2

matrix has a direct geometric meaning: the signed area of the parallelogram spanned by its columns.

\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc

Three cases:

• $\det A > 0$ — orientation preserved. The unit square stays oriented counterclockwise after transformation. Area scales by

\det A

.
• $\det A < 0$ — orientation reversed. The unit square becomes clockwise; the transformation includes a reflection somewhere.
• $\det A = 0$ — the columns are linearly dependent. The parallelogram has zero area; the transformation collapses the plane to a line or a point.

In the visualizer this is encoded in color: purple fill for positive determinant, pink for negative, gray for singular. For more see determinant of a matrix, signed area, and orientation in linear algebra.

Eigenvalues, Rank, and Matrix Types

Two more invariants pin down the structure of any

2 \times 2

matrix: the eigenvalues and the rank.

Eigenvalues are the roots of the characteristic polynomial

\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0

. They tell you what scaling factors

A

applies along its preserved directions. Real eigenvalues correspond to invariant lines visible as magenta dashed lines on the canvas. Complex eigenvalues come in conjugate pairs and indicate a rotational component with no real invariant line.

Rank is the dimension of the image: 2 for invertible matrices, 1 for matrices that collapse the plane to a line, 0 only for the zero matrix.

Together with the determinant, these classify the transformation:

• Rotation —

\det = 1

, complex eigenvalues on the unit circle.
• Reflection —

\det = -1

, eigenvalues

+1

and

-1

.
• Shear —

\det = 1

, repeated real eigenvalue, only one independent eigenvector.
• Scaling — diagonal, eigenvalues are the diagonal entries.
• Projection — rank 1, eigenvalues 0 and (nonzero).

For comprehensive treatment see eigenvalues and eigenvectors, matrix rank, and characteristic polynomial.

Linear Transformation

Identity

AMatrix

04Livet = 0.000

Key Terms

Getting Started

The Animation Slider

Preset Scenarios &mdash; Three Rank Categories

The Identity: the Transformation That Does Nothing

Full Rank: Reshaping Without Losing Anything

Rank 1: Collapsing the Plane Onto a Line

Rank 0: the Zero Map

Editing the Matrix A Directly

Display Layer Toggles

Reading the Live Card

The Classification Readout

What Is a Linear Transformation?

Determinant: Area and Orientation

Eigenvalues, Rank, and Matrix Types

Related Concepts

Preset Scenarios — Three Rank Categories