Matrices

Rectangular arrays of numbers that encode systems of equations, transformations, and data — the central objects of linear algebra.

A matrix — a rectangular array of numbers

2×3 matrix (2 rows, 3 columns)

[

123456

]

A₂₃

Entry notation

[

a₁₁a₁₂a₁₃a₂₁a₂₂a₂₃

]

aᵢⱼ = row i, col j

3×3 identity I

[

100010001

]

AI = IA = A

Square matrix

rows = columns

Zero matrix

all entries = 0

Diagonal matrix

off-diag = 0

Definition

A matrix is a rectangular array of numbers arranged in rows and columns. An $m \times n$ matrix has $m$ rows and $n$ columns.

A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}

The entry in row $i$ , column $j$ is written $a_{ij}$ (or $A_{ij}$ , or $(A)_{ij}$ ).

Special matrices:

Square matrix: $m = n$ (same number of rows and columns)
Identity matrix $I_n$ : $n \times n$ matrix with 1s on the diagonal, 0s elsewhere
Zero matrix $O$ : all entries are 0

Reading matrix entries

$A = \begin{pmatrix} 3 & -1 & 0 \\ 2 & 5 & -4 \end{pmatrix}$

This is a $2 \times 3$ matrix. $a_{11} = 3$ , $a_{12} = -1$ , $a_{23} = -4$ . Note the convention: row first, then column.

Try it

Write the $3 \times 3$ identity matrix $I_3$ . What happens when you multiply any vector $\mathbf{v}$ by $I_3$ ?

Solution

$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Multiplying $I_3 \mathbf{v} = \mathbf{v}$ — the identity matrix leaves every vector unchanged. It is the matrix analog of the number 1.

Related concepts