Textbook
A matrix $A\in\mathbb R^{m\times n}$ is a rectangular table of real numbers with $m$ rows and $n$ columns. Here $m,n$ are positive integers. Entry $A_{ij}$ lies in row $i$ and column $j$; the examples use one-based indices.
Its shape $m\times n$ is part of the object’s type. Equal matrices have the same shape and equal corresponding entries. Here $A$ names a table rather than a set: the context assigns its role. A row and a column containing the same numbers have different shapes and play different roles in multiplication.
For an $m\times n$ matrix, valid indices satisfy $1\le i\le m$ and $1\le j\le n$. An index outside those ranges selects no entry of this matrix. Zeros do not reduce its shape: zero rows and columns remain part of the table. Matrices of the same shape can differ in one entry; the same collection of numbers in different positions does not establish equality.
Definitions and further reading: Mathematics for Machine Learning.
In plain language
A matrix resembles a seating plan: an address requires the row first and the column second. Exchanging indices can select a different cell, and equal seat counts do not make two-by-three and three-by-two shapes identical. Zero occupies a full cell without shifting the remaining entries. Equal values do not merge; two tickets at the same price do not turn their seats into a shared chair.
Example
The illustrative matrix has first row $(1,2)$ and second row $(0,1)$. Its shape is followed by two different entries.
Matrices: Tables With Two Indices
Shape
Two rows and two columns give shape 2 × 2.
Step 1 of 3: Shape
Read all steps
Shape
Two rows and two columns give shape 2 × 2.
- A₁
- 1current2current
- A₂
- 0current1current
Row one, column two
Indices 1, 2 select the value 2.
- A₁
- 1current2 ←current
- A₂
- 0current1current
Row two, column one
Swapping the indices selects a different entry: 0.
- A₁
- 1current2current
- A₂
- 0 ←current1current
The example’s second row begins with zero, so $A_{21}=0$ identifies an actual entry rather than a missing cell. Both rows contain two entries, leaving the shape $2\times2$. Reading $A_{12}$ selects the first row and then its second position. State that order before calculating: an addressing mistake can produce a plausible number and survive every subsequent arithmetic step.
Related concepts: Matrix multiplication.