What does it mean to compose two matrices?

Composing two matrices means applying one transformation after another. If B sends v to Bv and A then sends that to A(Bv), the combined effect is a single linear transformation whose matrix is the product AB. Matrix multiplication is defined precisely so that (AB)v equals A(Bv) for every vector v.

Why is matrix multiplication not commutative?

Because doing two things in different orders need not give the same result. Shearing the plane and then rotating it produces a different picture from rotating and then shearing, so the matrices AB and BA are different. Some special pairs do commute, such as two rotations or two diagonal matrices, but in general the order matters.

In the product ABv, which matrix is applied first?

The matrix nearest the vector, B. Products are read right to left: ABv means first apply B to v, then apply A to the result. The tool's AB mode animates B in the first half and A in the second half, and the BA mode reverses that.

Which pairs of matrices commute?

Rotations of the plane always commute with each other, because composing rotations adds their angles. Diagonal matrices commute because they scale the axes independently. Any multiple of the identity commutes with every matrix, and powers of a single matrix commute with each other. In general, two matrices commute exactly when they can be diagonalized by the same basis of eigenvectors.

Why do two reflections compose to a rotation?

Each reflection preserves lengths but reverses orientation, with determinant −1. Two reversals cancel, so the composition preserves lengths and orientation, which makes it a rotation. Its angle is twice the angle between the two mirror lines, and swapping the order of the reflections gives the rotation in the opposite direction.

Matrix Composition

Linear Algebra·Matrix compositionTwo matrices stacked. A applied to (B applied to v) equals the single matrix AB applied to v. That's why matrix multiplication is defined the way it is.

v\u2080=(1.5, 1)start

Bv=(2, 1)after B

ABv=(0.71, 2.12)after AB

▸Composition I → B → ABt = 0.000phase 1 · applying B

apply Bapply A

M = B

t = 0.000

Shear, then rotate

non-commutative

First B shears horizontally, then A rotates 45°. The reverse order produces a different shape.
Learn more about non-commuting pairs · when matrices commute

Insight

Shears and rotations deform the plane in incompatible ways. AB and BA differ both in the final position of every vector and in the matrix entries.

AMatricesclick to edit

auto-computed

AB=

0.71-0.350.711.06

04LiveAB vs BA

matrices side by side

0.71-0.350.711.06

1.06-0.350.710.71

endpoint comparison

ABv(0.71, 2.12)BAv(1.24, 1.77)|ABv − BAv|0.64

≠A and B do NOT commute

Key Terms

Getting Started

Trail and Stages Views

The Two-Phase Animation

Preset Scenarios

A Commuting Pair: Two Rotations

A Non-Commuting Pair: Shear and Rotation

A Reveal: Two Reflections Make a Rotation

A Reveal: A Shear and Its Inverse

The Matrices Card and the Live Card

Display Layer Toggles

What Composition Is

When Matrices Commute

Key Properties

Related Concepts

Key Terms

Composition — applying one transformation after another. Applying

B

and then

A

sends

\mathbf{v}

A(B\mathbf{v})

.

Matrix product — the single matrix

AB

with

(AB)\mathbf{v} = A(B\mathbf{v})

for every

\mathbf{v}

. Matrix multiplication is defined so that this works.

Order of application — in

AB\mathbf{v}

the matrix nearest

\mathbf{v}

acts first:

B

, then

A

. Products read right to left.

Non-commutativity —

AB \neq BA

in general; the two orders are different transformations with different matrices.

Commuting pair — matrices with

AB = BA

, such as two rotations, two diagonal matrices, or anything paired with a multiple of the identity.

Trail — the two-segment path

\mathbf{v}_0 \to B\mathbf{v} \to AB\mathbf{v}

that a vector follows through the pipeline.

Stages — the same pipeline shown as three canvases: the input, the intermediate, and the final state.

Determinant of a product —

\det(AB) = \det A \cdot \det B

, the same in either order even when the matrices differ.

Getting Started

The tool shows a vector

\mathbf{v}

travelling through two matrices. In Trail view a single canvas draws the path

\mathbf{v}_0 \to B\mathbf{v} \to AB\mathbf{v}

in two colours, teal for the first matrix and purple for the second, with the alternative order

BA

drawn dashed for comparison. In Stages view the same pipeline is laid out as three canvases with chevrons between them.

Try this sequence to build intuition:

• Start on the Shear × rotate preset and press play. In the first half of the animation the grid shears and

\mathbf{v}

slides to

B\mathbf{v}

; in the second half the sheared plane rotates and

\mathbf{v}

arrives at

AB\mathbf{v}

.
• Flip the AB / BA toggle in the top bar. The same two matrices, applied in the other order, land

\mathbf{v}

somewhere else, and the dashed path shows where the first order went.
• Switch to Two rotations and flip the toggle again. Now both orders land in the same place: rotations commute.
• Drag

\mathbf{v}

in either view. The trail and both endpoints follow it.

The point: matrix multiplication is composition of transformations, and composition depends on order.

Trail and Stages Views

The top bar switches between two layouts of the same state.

• Trail — one canvas. The grid and unit square carry the partially applied product as the animation runs, and the vector's path is drawn as a breadcrumb trail with ghost markers at

\mathbf{v}_0

, the intermediate point, and the final point. The other order's path is drawn dashed behind it
• Stages — three canvases in a row: the input plane with

\mathbf{v}

, the plane after the first matrix with its unit square tinted in that matrix's colour, and the plane after both with the final square in indigo. Chevrons between them name the matrix applied at each step

Switching views changes nothing about the matrices, the vector, the mode or the animation position. Trail is better for seeing where a vector goes; Stages is better for seeing what happens to the whole plane at each step.

The Two-Phase Animation

The animation parameter

t

runs from

0

1

in two halves.

• For

t

from

0

0.5

, the first matrix is blended in from the identity: the grid morphs from square to its image under

B

(in

AB

mode), and

\mathbf{v}

slides in a straight line from

\mathbf{v}_0

B\mathbf{v}

• For

t

from

0.5

1

, the second matrix is blended in on top of the completed first: the grid morphs from

B

AB

, and

\mathbf{v}

slides from

B\mathbf{v}

AB\mathbf{v}

• The progress bar is split at the midpoint and coloured to match; the transport buttons step, scrub and reset

Because each phase blends one matrix linearly, the vector's path within a phase is a straight segment, which is why the trail is a two-segment polyline with a bend at the intermediate point. Selecting a preset or flipping the order restarts the animation from

t = 0

and plays it.

Preset Scenarios

The sidebar offers nine pairs in three groups.

• Commutative — two rotations, two diagonal scales, and a uniform scale with a rotation; in each case

AB = BA

and the dashed path lands exactly on the solid one
• Non-commutative — shear with rotation, two perpendicular shears, projection with rotation, and reflection with rotation; the two orders diverge, sometimes drastically
• Reveals — two reflections that compose to a rotation, and a shear with its inverse that composes to the identity

The four sections that follow freeze one representative of each situation, with the numbers the tool computes for the default vector. The explanation card in the tool links back to the matching section for whichever preset is active.

A Commuting Pair: Two Rotations

On the two-rotations preset

A

rotates by

60°

and

B

30°

. Applying

B

then

A

rotates by

90°

in total; applying

A

then

B

also rotates by

90°

. The two orders agree.

The frozen picture below is the end of the

AB

run: the solid trail bends at

B\mathbf{v}

and ends at

AB\mathbf{v}

, and the dashed

BA

trail, though it bends at a different intermediate point

A\mathbf{v}

, ends at exactly the same place.

Two rotations, end of the AB run

The solid trail bends at Bv = (0.8, 1.62) and ends at ABv = (-1, 1.5); the dashed BA trail bends elsewhere, at Av = (-0.12, 1.8), but ends at the same point. 30° + 60° = 60° + 30°.

The intermediate points differ,

B\mathbf{v}

\mathbf{v}

turned by

30°

while

A\mathbf{v}

\mathbf{v}

turned by

60°

, but the destinations coincide because

30° + 60° = 60° + 30°

. Rotations of the plane form a commutative group.

Commuting is a property of the pair, not of either matrix alone. Each of these rotations fails to commute with a shear, as the non-commutative presets show.

A Non-Commuting Pair: Shear and Rotation

On the default preset

B

shears horizontally by

0.5

and

A

rotates by

45°

. Shearing first and then rotating lands

\mathbf{v}

in one place; rotating first and then shearing lands it somewhere else.

The frozen picture below is the end of the

AB

run: the solid trail goes

\mathbf{v}_0 \to B\mathbf{v} \to AB\mathbf{v}

, the dashed trail goes

\mathbf{v}_0 \to A\mathbf{v} \to BA\mathbf{v}

, and the two endpoints are visibly apart.

Shear then rotate, end of the AB run

Shear first, then rotate: ABv = (0.71, 2.12). Rotate first, then shear: BAv = (1.24, 1.77). Two different destinations from the same two matrices; both products have determinant 1.

A shear stretches the plane along one direction while a rotation turns every direction, and the two deform the plane in ways that do not slide past each other. The matrices

AB

and

BA

have different entries, not just different pictures.

What they share is the determinant: both are

1

, since the shear and the rotation each preserve area. The unit square ends up as a parallelogram of area

1

in both orders, but a differently oriented one.

A Reveal: Two Reflections Make a Rotation

On the two-reflections preset

B

reflects across the x-axis and

A

reflects across

y = x

. Neither is a rotation, but their composition is:

AB

is the rotation by

90°

.

The frozen picture below shows

\mathbf{v}

arriving at

AB\mathbf{v}

, which is

\mathbf{v}_0

turned a quarter turn counterclockwise, and the unit square rotated but not flipped.

Two reflections, end of the AB run

v₀ = (1.5, 1) has arrived at ABv = (-1, 1.5), a quarter turn counterclockwise, and the unit square is rotated but not flipped: two orientation reversals cancel into a rotation by 90°.

Each reflection reverses orientation, with determinant

-1

, and two reversals cancel:

\det(AB) = (-1)(-1) = +1

. The composition preserves lengths, angles and orientation, which is the definition of a rotation, and its angle is twice the

45°

between the two mirrors.

The other order is the rotation by

-90°

. So this pair does not commute either, but the two results are as closely related as non-commuting products can be: inverses of each other.

A Reveal: A Shear and Its Inverse

On the shear-and-inverse preset

B

shears by

0.6

and

A

shears by

-0.6

. Applying one after the other undoes the first:

AB = I

, and

\mathbf{v}

returns to

\mathbf{v}_0

.

The frozen picture below shows the trail leaving

\mathbf{v}_0

for

B\mathbf{v}

and coming straight back, with the final unit square sitting exactly on the original.

Shear and its inverse, end of the AB run

The trail leaves v₀ for Bv = (2.1, 1) and comes straight back: ABv = (1.5, 1). AB = I, and the final unit square sits exactly on the original.

This is what an inverse matrix means as a transformation: the map that composes with the original to give the identity. It works in both orders here,

AB = BA = I

, which is always true of a matrix and its inverse.

The animation makes the undoing visible. The first half shears the plane; the second half shears it back, and at

t = 1

the grid is square again. Nothing has been lost, because a shear has determinant

1

and is invertible. Contrast the projection presets, where the first half flattens the plane and no second matrix can ever restore it.

The Matrices Card and the Live Card

The matrices card shows

A

and

B

as editable

2 \times 2

grids, with the product for the current order computed beside them. Editing an entry clears the preset and rebuilds everything.

The live card compares the two orders directly:

•

AB

and

BA

as matrices, side by side
•

AB\mathbf{v}

and

BA\mathbf{v}

for the current

\mathbf{v}

• A verdict, commute or do not commute, from whether the two products agree
•

\det(AB)

, which always equals

\det A \cdot \det B

and is therefore the same for both orders

The determinant line is the reminder that some things about a product do not depend on order: area scaling and orientation are multiplicative, even though the shapes themselves differ.

Display Layer Toggles

Six chips switch the overlays on and off.

• grid — the morphing background grid
• unit sq — the unit square under the current partial product, tinted by stage
• ghosts — the markers at

\mathbf{v}_0

, the intermediate point and the endpoint, which fill in as the animation passes them
• AB path — the primary trail in the current order's colours
• BA path — the dashed alternative trail and its ghosts
• labels — the names on every marker

Turning off the grid and unit square leaves only the two trails, which is the clearest way to compare where the two orders send a single vector.

What Composition Is

B

sends

\mathbf{v}

B\mathbf{v}

and

A

then sends that to

A(B\mathbf{v})

, the overall effect is a single linear transformation, and its matrix is the product

AB

A(B\mathbf{v}) = (AB)\,\mathbf{v} \quad \text{for every } \mathbf{v}

This is not a coincidence about matrices; it is the reason matrix multiplication is defined the way it is. The column

j

AB

A

applied to column

j

B

, which is where the row-times-column rule comes from: the entry

(AB)_{ij} = \sum_k A_{ik} B_{kj}

is exactly what falls out of applying

A

to the image of the

j

-th basis vector under

B

.

Two consequences follow immediately. Products read right to left, since the matrix nearest the vector acts first. And products need not commute, because doing two things in different orders need not give the same result: shearing a picture and then rotating it is not the same as rotating and then shearing.

For the arithmetic of the product itself, see the matrix multiplication visualizer; for transformations as matrices in general, see the matrix representation theory page.

When Matrices Commute

Most pairs of matrices do not commute, but several important families do.

• Rotations of the plane — composing rotations adds their angles, and addition is commutative:

R(\alpha)R(\beta) = R(\alpha + \beta) = R(\beta)R(\alpha)

• Diagonal matrices — they scale the axes independently, so the order of two independent scalings does not matter
• Multiples of the identity —

kI

commutes with every matrix, since

kI \cdot M = kM = M \cdot kI

• Powers of one matrix —

M^a

and

M^b

always commute
• Simultaneously diagonalizable matrices — matrices that share a full set of eigenvectors commute, and this is the general criterion

The test in the tool is direct: compute

AB

and

BA

and compare. When they agree, the dashed alternative path lands on the solid one and the live card says so. Note that

\det(AB) = \det(BA)

always, so the determinant cannot tell the two orders apart; only the matrices themselves can.

Key Properties

Composition obeys a few rules that hold regardless of order.

• Associativity:

(AB)C = A(BC)

; three transformations in sequence compose the same way however the pairs are grouped
• Identity:

AI = IA = A

; the identity matrix is the do-nothing transformation
• Inverse:

AA^{-1} = A^{-1}A = I

; a transformation followed by its undoing is the identity, in either order
• Determinant:

\det(AB) = \det A \cdot \det B

• Inverse of a product:

(AB)^{-1} = B^{-1}A^{-1}

; to undo

B

-then-

A

, undo

A

first, then

B

• Transpose of a product:

(AB)^T = B^T A^T

• Rank:

\operatorname{rank}(AB) \leq \min(\operatorname{rank} A, \operatorname{rank} B)

; composing with a rank-

1

projection can never restore a lost dimension
• Not commutative:

AB \neq BA

in general, and

AB = BA

is a special property of the pair