How do you add two matrices?

Two matrices can be added only if they share the same dimensions. The sum is computed entry by entry: the (i,j) entry of A + B equals the sum of the (i,j) entries of A and B. Matrix addition is commutative and associative, with the zero matrix serving as the additive identity.

How does matrix multiplication work?

To multiply an m × n matrix A by an n × p matrix B, the number of columns of A must equal the number of rows of B. Each entry of the product AB is the dot product of a row of A with a column of B, producing an m × p result. Matrix multiplication is associative and distributive but not commutative.

Why is matrix multiplication not commutative?

Matrix multiplication is not commutative because each entry depends on the order of row-column pairings. Even for two square matrices of the same size, AB and BA generally produce different results. Additionally, the product of nonzero matrices can be zero, and cancellation does not hold unless the left factor is invertible.

What does transposing a matrix do?

Transposing a matrix converts its rows into columns and vice versa, turning an m × n matrix into an n × m matrix. The transpose reverses the order of products: (AB)ᵀ = BᵀAᵀ. A matrix equal to its own transpose is called symmetric.

What are elementary matrices used for?

An elementary matrix results from performing a single row operation on the identity matrix. Left-multiplying any matrix by an elementary matrix performs that same row operation. Every invertible matrix can be expressed as a product of elementary matrices, connecting row reduction to matrix factorization.

Operations on Matrices

Matrix Addition

Matrix Subtraction

Scalar Multiplication

Linear Combinations of Matrices

Matrix Multiplication — Definition

Matrix Multiplication — Properties

Matrix Multiplication — Column and Row Interpretations

The Transpose

Matrix Powers

Elementary Matrices

Matrix Decompositions

Dimension Summary

Manipulating Matrices

Matrices support a family of operations — addition, scalar multiplication, matrix multiplication, transposition, and exponentiation — each with its own rules and dimension requirements. Matrix multiplication stands apart from the rest: it is not commutative, it demands compatible dimensions, and it admits several geometric and algebraic interpretations that make it one of the richest operations in all of mathematics.

Matrix Addition

Two matrices of the same size can be added entry by entry. If

A

and

B

are both

m \times n

, their sum is the

m \times n

matrix with entries

(A + B)_{ij} = a_{ij} + b_{ij}

For example,

\begin{pmatrix} 1 & 4 \\ -2 & 3 \\ 0 & 5 \end{pmatrix} + \begin{pmatrix} 3 & -1 \\ 6 & 0 \\ 2 & -4 \end{pmatrix} = \begin{pmatrix} 4 & 3 \\ 4 & 3 \\ 2 & 1 \end{pmatrix}

If the dimensions do not match, the sum is undefined — there is no way to add a

2 \times 3

matrix to a

3 \times 2

matrix.

Addition is commutative (

A + B = B + A

) and associative (

(A + B) + C = A + (B + C)

). The zero matrix

O

of the same size serves as the additive identity (

A + O = A

), and the additive inverse of

A

-A = (-a_{ij})

, so

A + (-A) = O

Matrix Subtraction

Subtraction is defined as addition of the negative:

A - B = A + (-B)

Entry by entry,

(A - B)_{ij} = a_{ij} - b_{ij}

. The same dimension requirement applies — both matrices must have identical shapes. There is nothing deeper here than combining addition and negation, but it appears often enough to warrant its own notation.

Scalar Multiplication

Multiplying a matrix by a scalar

c

scales every entry:

(cA)_{ij} = c \cdot a_{ij}

For example,

-2 \begin{pmatrix} 1 & 3 & -4 \\ 0 & 5 & 2 \end{pmatrix} = \begin{pmatrix} -2 & -6 & 8 \\ 0 & -10 & -4 \end{pmatrix}

Scalar multiplication distributes over matrix addition (

c(A + B) = cA + cB

), distributes over scalar addition (

(c + d)A = cA + dA

), associates with itself (

c(dA) = (cd)A

), and has

1

as its identity (

1 \cdot A = A

). Multiplying by

0

produces the zero matrix.

Linear Combinations of Matrices

Given matrices

A_1, A_2, \dots, A_k

of the same size and scalars

c_1, c_2, \dots, c_k

, the expression

c_1 A_1 + c_2 A_2 + \cdots + c_k A_k

is a linear combination of matrices. Addition and scalar multiplication together give the set of all

m \times n

matrices the structure of a vector space. The dimension of this space is

mn

— one degree of freedom for each entry. The standard basis consists of the

mn

matrices that have a single

1

in one position and zeros everywhere else.

Matrix Multiplication — Definition

For

A

of size

m \times n

and

B

of size

n \times p

, the product

AB

is an

m \times p

matrix whose

(i,j)

entry is the dot product of row

i

A

with column

j

B

(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}

The number of columns of

A

must equal the number of rows of

B

. If this compatibility condition fails, the product is undefined.

Worked Example

\begin{pmatrix} 1 & 0 & 3 \\ 2 & -1 & 4 \end{pmatrix} \begin{pmatrix} 5 & 1 \\ 2 & -3 \\ 0 & 6 \end{pmatrix}

The left matrix is

2 \times 3

and the right is

3 \times 2

, so the product is

2 \times 2

. Computing each entry:

(1)(5) + (0)(2) + (3)(0) = 5, \quad (1)(1) + (0)(-3) + (3)(6) = 19

(2)(5) + (-1)(2) + (4)(0) = 8, \quad (2)(1) + (-1)(-3) + (4)(6) = 29

AB = \begin{pmatrix} 5 & 19 \\ 8 & 29 \end{pmatrix}

Each entry required

n = 3

multiplications and

n - 1 = 2

additions. The full product required

m \times p = 4

such computations.

Matrix Multiplication — Properties

Matrix multiplication obeys several familiar algebraic rules and violates one that is deeply ingrained from scalar arithmetic.

Associativity holds:

(AB)C = A(BC)

whenever all products are defined. Distribution holds on both sides:

A(B + C) = AB + AC

and

(A + B)C = AC + BC

. Scalars pass through freely:

c(AB) = (cA)B = A(cB)

. The identity matrix satisfies

AI = IA = A

whenever the dimensions are compatible.

Commutativity, however, fails. In general,

AB \neq BA

, even when both products happen to be defined. For a concrete counterexample, take

A = \begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix}

and

B = \begin{pmatrix} 0 & 0 \\ 3 & 4 \end{pmatrix}

. Then

AB = \begin{pmatrix} 6 & 8 \\ 0 & 0 \end{pmatrix}

while

BA = \begin{pmatrix} 0 & 0 \\ 3 & 6 \end{pmatrix}

.

Two further properties distinguish matrix multiplication from scalar multiplication. The product of two nonzero matrices can be zero: if

A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}

and

B = \begin{pmatrix} 2 & -4 \\ -1 & 2 \end{pmatrix}

, then

AB = O

even though neither

A

nor

B

is zero. Cancellation also fails:

AB = AC

does not imply

B = C

unless

A

is invertible.

Property	Status	Identity / note
Associativity	holds	(AB)C = A(BC)
Left distribution	holds	A(B + C) = AB + AC
Right distribution	holds	(A + B)C = AC + BC
Scalar passage	holds	c(AB) = (cA)B = A(cB)
Identity	holds	AI = IA = A
Commutativity	fails	AB ≠ BA in general
Zero-product law	fails	AB = O can occur with A ≠ O and B ≠ O
Cancellation	fails	AB = AC ⇏ B = C unless A is invertible

Matrix Multiplication — Column and Row Interpretations

The entry-by-entry formula is the most common way to define matrix multiplication, but two alternative viewpoints often provide sharper insight.

The column interpretation says that column

j

AB

is obtained by multiplying

A

times column

j

B

AB = \begin{pmatrix} A\mathbf{b}_1 & A\mathbf{b}_2 & \cdots & A\mathbf{b}_p \end{pmatrix}

Each column of the product is a linear combination of the columns of

A

, with weights given by the corresponding column of

B

. This is the view that connects matrix multiplication to linear transformations: the product

AB

applies the transformation

A

to each column of

B

independently.

The row interpretation says that row

i

AB

equals row

i

A

times the entire matrix

B

. Each row of the product is a linear combination of the rows of

B

, weighted by the entries in the corresponding row of

A

.

A third perspective writes the product as a sum of rank-one outer products:

AB = \sum_{k=1}^{n} (\text{column } k \text{ of } A)(\text{row } k \text{ of } B)

Each term is an

m \times p

matrix of rank at most one, and their sum is the full product. This decomposition appears in low-rank approximation theory and in the analysis of the singular value decomposition.

Interpretation	Building block	Formula for AB	When it shines
Entry-by-entry	scalar (i, j) entries	(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ	hand computation, definition
Column view	columns of B	column j of AB = A · bⱼ	link to linear transformations
Row view	rows of A	row i of AB = aᵢ · B	rank, row reduction
Outer product sum	rank-1 matrices	AB = Σₖ (col k of A)(row k of B)	low-rank approximation, SVD

The Transpose

The transpose of an

m \times n

matrix

A

is the

n \times m

matrix

A^T

obtained by converting rows into columns:

(A^T)_{ij} = a_{ji}

For example,

A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \quad \Longrightarrow \quad A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}

The transpose satisfies

(A^T)^T = A

, distributes over addition (

(A + B)^T = A^T + B^T

), and commutes with scalar multiplication (

(cA)^T = cA^T

). The product rule reverses the order:

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

This reversal is a frequent source of errors and is worth memorizing as a pattern: transposing a product is like reading it backward.

A matrix satisfying

A = A^T

is called symmetric. For any matrix

A

of any shape, the products

A^T A

and

AA^T

are both symmetric — this is immediate from the product rule, since

(A^T A)^T = A^T (A^T)^T = A^T A

Matrix Powers

For a square matrix

A

, powers are defined by repeated multiplication:

A^0 = I, \quad A^1 = A, \quad A^k = \underbrace{A \cdot A \cdots A}_{k \text{ factors}}

The usual exponent laws hold:

A^j A^k = A^{j+k}

and

(A^j)^k = A^{jk}

. When

A

is invertible, negative powers are defined as

A^{-k} = (A^{-1})^k

, extending the exponent laws to all integers.

One rule from scalar arithmetic does not carry over. Since matrix multiplication is not commutative, the identity

(AB)^k = A^k B^k

is false in general. Expanding

(AB)^2 = ABAB

, there is no way to rearrange this into

A^2 B^2 = AABB

without commutativity.

Powers of specific matrix types are particularly well-behaved. For a diagonal matrix

D = \text{diag}(d_1, \dots, d_n)

, the

k

-th power is

D^k = \text{diag}(d_1^k, \dots, d_n^k)

— each diagonal entry is raised to the

k

-th power independently. This simplicity is one of the main reasons diagonalization is so useful: writing

A = PDP^{-1}

gives

A^k = PD^kP^{-1}

, reducing an expensive matrix power to a cheap diagonal power.

Elementary Matrices

An elementary matrix is the result of performing a single row operation on the identity matrix. There are three types, corresponding to the three row operations: swapping two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another.

The key property is that left-multiplying a matrix

A

by an elementary matrix

E

performs the corresponding row operation on

A

. If

E

swaps rows

2

and

3

of the identity, then

EA

swaps rows

2

and

3

A

. If

E

scales row

1

of the identity by

5

, then

EA

scales row

1

A

5

.

Every elementary matrix is invertible, and its inverse is another elementary matrix of the same type: the inverse of a row swap is the same row swap, the inverse of scaling by

k

is scaling by

1/k

, and the inverse of adding

c

times row

i

to row

j

is subtracting

c

times row

i

from row

j

.

This leads to a structural result: every invertible matrix can be written as a product of elementary matrices. Since Gaussian elimination reduces an invertible matrix to the identity through a sequence of row operations, each operation corresponds to an elementary matrix, and reversing the sequence expresses the original matrix as their product. This factorization is more conceptual than computational, but it underpins the theoretical foundations of the determinant and the inverse.

Matrix Decompositions

A matrix decomposition (or factorization) expresses a matrix as a product of simpler matrices with known structure. Decompositions are among the most powerful tools in computational linear algebra, converting hard problems into sequences of easy ones.

The LU decomposition writes

A = LU

where

L

is lower triangular and

U

is upper triangular. It captures the essence of Gaussian elimination in matrix form and makes solving linear systems with multiple right-hand sides efficient: once

L

and

U

are known, each system reduces to two triangular solves.

The QR decomposition writes

A = QR

where

Q

is orthogonal and

R

is upper triangular. It is the foundation of least-squares computation and several eigenvalue algorithms.

The Cholesky decomposition writes

A = LL^T

for symmetric positive definite matrices, achieving the work of LU in roughly half the computation by exploiting symmetry.

The eigendecomposition writes

A = PDP^{-1}

where

D

is diagonal, placing the eigenvalues on the diagonal and the eigenvectors in the columns of

P

. It applies only to diagonalizable matrices.

The singular value decomposition writes

A = U\Sigma V^T

where

U

and

V

are orthogonal and

\Sigma

is diagonal with nonnegative entries. Unlike the eigendecomposition, the SVD exists for every matrix of every shape. It reveals the rank, the fundamental subspaces, and the best low-rank approximation to

A

, making it one of the most broadly applicable tools in the subject.

Each of these decompositions has its own page with full derivations and worked examples.

Decomposition	Form	Requires	Primary use
LU	A = LU (L lower, U upper triangular)	any square matrix (with pivoting)	solving Ax = b for many right-hand sides
QR	A = QR (Q orthogonal, R upper triangular)	any matrix	least squares, eigenvalue algorithms
Cholesky	A = LLᵀ	symmetric positive definite	half-cost LU for SPD systems
Eigendecomposition	A = PDP⁻¹ (D diagonal)	diagonalizable square matrix	matrix powers, spectral analysis
SVD	A = UΣVᵀ (U, V orthogonal; Σ diagonal ≥ 0)	any matrix of any shape	rank, fundamental subspaces, best low-rank approximation

Dimension Summary

The dimension requirements for each operation are worth collecting in one place. Addition and subtraction require both matrices to share the same

m \times n

dimensions, and the result is

m \times n

. Scalar multiplication imposes no restriction and preserves the original dimensions. The product

AB

requires the column count of

A

to match the row count of

B

— if

A

m \times n

and

B

n \times p

, the result is

m \times p

. The transpose of an

m \times n

matrix is

n \times m

. Powers

A^k

require

A

to be square and produce a matrix of the same size.

A common source of confusion is the product rule. The product of two

n \times n

matrices is

n \times n

, but the product of a

2 \times 3

matrix with a

3 \times 5

matrix is

2 \times 5

. The "inner" dimensions must match and are consumed; the "outer" dimensions survive into the result.

Operation	Input requirement	Result size	Key property
Addition / subtraction	A and B both m × n	m × n	commutative; additive identity O
Scalar multiplication	A is m × n; c any scalar	m × n	distributes over both matrix and scalar sum
Linear combination	all Aᵢ same m × n shape	m × n	turns m × n matrices into a vector space of dimension mn
Matrix multiplication	A is m × n, B is n × p (inner dims match)	m × p	associative, NOT commutative; inner dim consumed
Transpose	A is m × n (no other restriction)	n × m	(Aᵀ)ᵀ = A; (AB)ᵀ = BᵀAᵀ (order reverses)
Power Aᵏ	A square (n × n), k ≥ 0 (or k ∈ ℤ if invertible)	n × n	Aʲ Aᵏ = Aʲ⁺ᵏ; (AB)ᵏ ≠ Aᵏ Bᵏ in general