1. What a matrix is

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.

Matrices are used to represent:

Linear transformations
Systems of equations
Data tables
Graph and network relationships
Recurrence relations and computational models

In linear algebra, matrices are most often studied as algebraic objects that encode linear maps.

Entry notation

If $A$ is a matrix, its entry in row $i$ and column $j$ is written as:

a_{ij}

For example:

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}

then:

a_{12} = 2,\quad a_{21} = 4

2. Matrix notation and dimensions

The size or dimension of a matrix is written as:

m \times n

where:

$m$ = number of rows
$n$ = number of columns

Examples:

\begin{bmatrix} 1 & 0 \\ 3 & 5 \end{bmatrix} \quad \text{is } 2 \times 2

\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \quad \text{is } 1 \times 3

\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \quad \text{is } 3 \times 1

Row and column vectors

A row vector is a $1 \times n$ matrix.

A column vector is an $m \times 1$ matrix.

Vectors are special cases of matrices.

Equality of matrices

Two matrices are equal if and only if:

They have the same dimensions
Every corresponding entry is equal

3. Types of matrices

Type	Definition	Example
Zero matrix	All entries are zero	$\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}$
Square matrix	Same number of rows and columns	$2 \times 2$, $3 \times 3$
Diagonal matrix	Nonzero entries only on the main diagonal	$\begin{bmatrix}2 & 0\\0 & -1\end{bmatrix}$
Scalar matrix	Diagonal matrix with equal diagonal entries	$cI$
Identity matrix	Ones on diagonal, zeros elsewhere	$I_3$
Upper triangular	Zeros below diagonal	$\begin{bmatrix}1 & 2\\0 & 3\end{bmatrix}$
Lower triangular	Zeros above diagonal	$\begin{bmatrix}1 & 0\\4 & 2\end{bmatrix}$
Symmetric	$A^T = A$	$\begin{bmatrix}1 & 2\\2 & 5\end{bmatrix}$
Skew-symmetric	$A^T = -A$	$\begin{bmatrix}0 & 3\\-3 & 0\end{bmatrix}$
Orthogonal	$A^T A = I$	Rotation matrices
Singular	Not invertible	Determinant $= 0$
Nonsingular	Invertible	Determinant $\ne 0$

Identity matrix

The identity matrix $I_n$ acts like the number 1 under matrix multiplication:

$$ AI_n = I_mA = A $$

when dimensions match.

Trace

For a square matrix, the trace is the sum of diagonal entries:

\operatorname{tr}(A) = \sum_{i=1}^{n} a_{ii}

Trace is defined only for square matrices.

4. Matrix operations

Addition and subtraction

Matrix addition and subtraction require equal dimensions and are done entrywise.

If $A = [a_{ij}]$ and $B = [b_{ij}]$, then:

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

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

Example:

\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8\\ 10 & 12 \end{bmatrix}

Scalar multiplication

For scalar $c$:

cA = [ca_{ij}]

This multiplies every entry by $c$.

Linear combination

A linear combination of matrices has the form:

c_1A_1 + c_2A_2 + \cdots + c_kA_k

This is useful in vector spaces of matrices.

Properties of addition and scalar multiplication

For matrices of compatible size:

$$ A+B = B+A $$

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

$$ A+0 = A $$

$$ A + (-A) = 0 $$

$$ c(A+B) = cA + cB $$

$$ (c+d)A = cA + dA $$

$$ (cd)A = c(dA) $$

5. Matrix multiplication

Matrix multiplication is not entrywise. It is defined by row-by-column products.

If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is defined and has size $m \times p$.

Compatibility rule

The number of columns of the left matrix must equal the number of rows of the right matrix.

If dimensions do not match, the product is undefined.

Product formula

The $(i,j)$ entry of $AB$ is:

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

Example:

\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 19 & 22\\ 43 & 50 \end{bmatrix}

because:

19 = 1\cdot 5 + 2\cdot 7

22 = 1\cdot 6 + 2\cdot 8

Key properties

Matrix multiplication is:

Associative: $(AB)C = A(BC)$
Distributive: $A(B+C) = AB + AC$
Compatible with scalars: $(cA)B = c(AB) = A(cB)$

Matrix multiplication is generally not commutative:

AB \ne BA

in most cases.

Powers of a matrix

For a square matrix:

A^2 = AA,\quad A^3 = AAA

and in general:

A^k = \underbrace{A\cdots A}_{k\text{ factors}}

Matrix powers are only defined for square matrices.

Useful interpretation

If $A$ is a matrix and $x$ is a vector, then $Ax$ is a linear combination of the columns of $A$ weighted by the components of $x$.

That viewpoint is often the fastest way to interpret $Ax=b$.

Interactive visual

Linear transformation

Change the entries of a 2x2 matrix to see how it stretches and shears a grid.

a 1.25 b 0.5 c -0.25 d 1

Determinant 1.375

Trace 2.25

6. Transpose and symmetry

The transpose of a matrix $A$ is denoted $A^T$ and is formed by swapping rows and columns.

If $A = [a_{ij}]$, then:

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

Example:

\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix}^T = \begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix}

Transpose rules

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

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

$$ (cA)^T = cA^T $$

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

The last identity reverses the order.

Symmetric matrices

A square matrix is symmetric if:

$$ A^T = A $$

Symmetric matrices have mirrored entries across the main diagonal.

Skew-symmetric matrices

A square matrix is skew-symmetric if:

$$ A^T = -A $$

Every diagonal entry of a skew-symmetric matrix is zero.

7. Row reduction and echelon form

Row operations are used to solve systems and simplify matrices.

Elementary row operations

Swap two rows
Multiply a row by a nonzero scalar
Replace a row by itself plus a multiple of another row

These operations preserve the solution set of a linear system.

Row echelon form

A matrix is in row echelon form if:

All zero rows are at the bottom
Each leading nonzero entry is to the right of the one above it
Entries below each leading entry are zero

Reduced row echelon form

A matrix is in reduced row echelon form if, in addition:

Every leading entry is 1
Each leading 1 is the only nonzero entry in its column

The reduced row echelon form is unique.

Pivots

The first nonzero entry in a nonzero row is a pivot.

Pivot columns identify which variables are basic in a linear system.

How to row-reduce efficiently

Work left to right, top to bottom.
Create a pivot in the current column.
Eliminate entries below the pivot.
Move to the next row and next column.
Back-substitute if you need reduced row echelon form.

8. Determinants

The determinant is a scalar associated with a square matrix.

It measures, among other things:

Scaling of area or volume under the linear transformation
Whether the matrix is invertible
Orientation change

Determinant of a 2 by 2 matrix

For:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

the determinant is:

\det(A) = ad - bc

Determinant of a 3 by 3 matrix

For:

A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}

one expansion is:

\det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)

Determinant properties

For square matrices:

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

\det(A^T) = \det(A)

\det(I) = 1

If $A$ is triangular, then:

\det(A) = \prod_{i=1}^{n} a_{ii}

Determinant and invertibility

A square matrix is invertible if and only if:

\det(A) \ne 0

If:

\det(A) = 0

then the matrix is singular.

Cofactor expansion

For a square matrix, the determinant can be expanded along a row or column using minors and cofactors.

This method is conceptually useful, but for large matrices row reduction is often faster.

9. Inverse matrices

If $A$ is square, its inverse $A^{-1}$ satisfies:

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

An inverse exists exactly when $A$ is nonsingular.

Why inverses matter

If $Ax=b$ and $A^{-1}$ exists, then:

x = A^{-1}b

This solves the system in one step, though computing the inverse directly is often not the best numerical method.

Inverse of a 2 by 2 matrix

For:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

if $ad-bc \ne 0$, then:

A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Inverse properties

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

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

(cA)^{-1} = \frac{1}{c}A^{-1}

for nonzero scalar $c$.

Practical note

To check whether your inverse is correct, multiply:

AA^{-1} = I

on one side and, if needed, on both sides.

10. Systems of linear equations

A linear system can be written as:

$$ Ax = b $$

where:

$A$ is the coefficient matrix
$x$ is the vector of unknowns
$b$ is the constant vector

Augmented matrix

The system is often represented as:

[A \mid b]

Row reduction is then used to determine whether the system has:

No solution
Exactly one solution
Infinitely many solutions

Interpret the row-reduced form

If a row reduces to:

[0\ 0\ \cdots\ 0 \mid c]

with $c \ne 0$, the system is inconsistent.

If every variable column has a pivot, the solution is unique.

If some variables are free, the system has infinitely many solutions.

Homogeneous systems

A homogeneous system has the form:

$$ Ax = 0 $$

It always has the trivial solution:

$$ x = 0 $$

Nontrivial solutions exist exactly when the columns of $A$ are linearly dependent.

Solution structure

For a consistent system, the general solution is:

$$ x = x_p + x_h $$

where:

$x_p$ is one particular solution
$x_h$ is the general solution of the homogeneous system

11. Rank, nullity, and subspaces

Rank

The rank of a matrix is the number of pivot columns, equivalently the dimension of the column space.

It is also the dimension of the row space.

Nullity

The nullity of a matrix is the dimension of its null space:

\mathcal{N}(A) = \{x : Ax = 0\}

Rank-nullity theorem

For an $m \times n$ matrix:

\operatorname{rank}(A) + \operatorname{nullity}(A) = n

This is one of the most useful bookkeeping identities in linear algebra.

Column space

The column space is the span of the columns of $A$.

It is the set of all vectors $b$ for which $Ax=b$ has at least one solution.

Row space

The row space is the span of the rows of $A$.

Row operations preserve the row space up to equivalence of the row span.

Null space

The null space contains all vectors mapped to zero by $A$.

If the null space contains only the zero vector, then $A$ has full column rank.

12. Eigenvalues and eigenvectors

For a square matrix $A$, a nonzero vector $v$ is an eigenvector if:

Av = \lambda v

for some scalar $\lambda$ called the eigenvalue.

Characteristic equation

Eigenvalues satisfy:

\det(A - \lambda I) = 0

This is called the characteristic equation.

How to find eigenvalues

Compute $A - \lambda I$.
Compute $\det(A - \lambda I)$.
Set the result equal to zero.
Solve for $\lambda$.

How to find eigenvectors

For each eigenvalue $\lambda$:

Form $A - \lambda I$.
Solve:

(A - \lambda I)v = 0

Any nonzero solution is an eigenvector.

Geometric interpretation

Eigenvectors are directions that are preserved by the linear transformation.

The transformation only scales or reverses them, rather than changing their direction.

Algebraic and geometric multiplicity

Algebraic multiplicity: multiplicity of an eigenvalue as a root of the characteristic polynomial
Geometric multiplicity: dimension of the eigenspace

Always:

1 \le \text{geometric multiplicity} \le \text{algebraic multiplicity}

13. Diagonalization and matrix powers

A matrix $A$ is diagonalizable if:

A = PDP^{-1}

where:

$D$ is diagonal
The columns of $P$ are eigenvectors of $A$
The diagonal entries of $D$ are the corresponding eigenvalues

Why diagonalization is useful

Once diagonalized, powers become easy:

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

Since $D$ is diagonal, $D^k$ is easy to compute by raising each diagonal entry to the $k$th power.

Diagonalization test

A square matrix is diagonalizable if it has enough linearly independent eigenvectors to form a basis.

In practice, this often means finding $n$ independent eigenvectors for an $n \times n$ matrix.

Symmetric matrices

Real symmetric matrices have especially nice behavior:

Their eigenvalues are real
Eigenvectors corresponding to distinct eigenvalues are orthogonal
They are orthogonally diagonalizable

For a symmetric matrix:

$$ A = QDQ^T $$

where $Q$ is orthogonal.

14. Special matrices and decompositions

Orthogonal matrices

A square matrix $Q$ is orthogonal if:

$$ Q^TQ = I $$

Equivalently:

Q^{-1} = Q^T

Orthogonal matrices preserve lengths and angles.

Projection matrices

A projection matrix satisfies:

$$ P^2 = P $$

It projects vectors onto a subspace.

Idempotent and nilpotent matrices

An idempotent matrix satisfies:

$$ A^2 = A $$

A nilpotent matrix satisfies:

$$ A^k = 0 $$

for some positive integer $k$.

LU decomposition

Many matrices can be factored as:

$$ A = LU $$

where:

$L$ is lower triangular
$U$ is upper triangular

This is useful for solving multiple systems with the same coefficient matrix.

QR decomposition

Another common factorization is:

$$ A = QR $$

where:

$Q$ has orthonormal columns
$R$ is upper triangular

QR decomposition is central in least-squares problems and numerical algorithms.

15. Problem-solving workflow

Use this sequence for most matrix problems.

Step 1: Identify the object

Decide whether you are working with:

A raw matrix
A linear system
A transformation
An eigenvalue problem
A decomposition problem

Step 2: Check dimensions

Verify compatibility before computing:

Addition requires equal size
Multiplication requires matching inner dimensions
Inverses require square matrices
Eigenvalues require square matrices

Step 3: Choose the right tool

Common choices:

Row reduction for systems and rank
Determinants for invertibility checks
Inverses for algebraic rearrangement
Eigenvalues for dynamics, stability, and diagonalization
Factorizations for efficient computation

Step 4: Simplify symbolically

Keep expressions symbolic until the structure is clear.

Do not substitute numbers before the matrix form is organized.

Step 5: Verify the result

Check:

Dimensions
Signs
Pivot structure
Invertibility conditions
Consistency of the system

For eigenvalue problems, verify with:

Av = \lambda v

For inverses, verify with:

AA^{-1} = I

16. Formula sheet

Core definitions

A = [a_{ij}]

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

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

Matrix sizes

If $A$ is $m \times n$ and $B$ is $n \times p$, then:

AB \text{ is } m \times p

Identity and inverse

$$ AI = IA = A $$

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

Determinants

For $2 \times 2$ matrices:

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

For triangular matrices:

\det(A) = \prod_{i=1}^{n} a_{ii}

Matrix is invertible if and only if:

\det(A) \ne 0

Row reduction

Elementary row operations:

Swap two rows
Multiply a row by a nonzero scalar
Add a multiple of one row to another

Rank-nullity

\operatorname{rank}(A) + \operatorname{nullity}(A) = n

Eigenvalues

Av = \lambda v

\det(A - \lambda I) = 0

Diagonalization

A = PDP^{-1}

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

Orthogonality

$$ Q^TQ = I $$

Q^{-1} = Q^T

17. Common mistakes to avoid

Treating matrix multiplication as commutative.
Adding or multiplying matrices with incompatible dimensions.
Forgetting that only square matrices can have determinants and inverses.
Confusing row operations with column operations.
Dropping the order reversal in $(AB)^T = B^TA^T$.
Using $\det(A) \ne 0$ as a shortcut without confirming $A$ is square.
Assuming every matrix is diagonalizable.
Mixing up eigenvalues and eigenvectors.
Forgetting that $Ax=b$ can have zero, one, or infinitely many solutions.
Misreading pivots and free variables after row reduction.
Treating a singular matrix as invertible.
Forgetting to verify solutions by substitution or matrix multiplication.

Sources

OpenStax Mathematics
Mathematics LibreTexts
Stewart, Calculus: Early Transcendentals
Lay, Linear Algebra and Its Applications
Rosen, Discrete Mathematics and Its Applications
Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems
Blitzstein and Hwang, Introduction to Probability
Parell GitHub repository