3.1 What is a Matrix?
Introduction
Imagine you run a store that sells three products: A, B, and C. You need to keep track of each product’s price, inventory, and sales figures. The most natural approach? Arrange the data in a table – rows for the products, columns for the attributes. At a glance, you can see everything.
In mathematics, this rectangular arrangement of numbers is called a matrix. But a matrix is far more than a data table. Write a system of equations in matrix form, and you can solve it systematically. Represent rotation, scaling, or reflection as a single matrix, and you can transform entire spaces with one multiplication. Matrices are the universal language of modern mathematics, physics, computer science, and machine learning.
Definition of a Matrix
Basic Definition
An $m \times n$ matrix is a rectangular array of numbers with $m$ rows and $n$ columns.
\[A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}\]The entry $a_{ij}$ sits in the $i$-th row and $j$-th column. A useful mnemonic: the row index always comes first, then the column index – just like how you say “Row $i$, Column $j$.”
Connection to Systems of Equations
Consider the following system of linear equations:
\[\begin{cases} 2x + 3y = 5 \\ x - y = 1 \end{cases}\]Collect the coefficients into a matrix:
\[A = \begin{pmatrix} 2 & 3 \\ 1 & -1 \end{pmatrix}, \quad \vec{x} = \begin{pmatrix} x \\ y \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}\]Now the entire system collapses into a single matrix equation:
\[A\vec{x} = \vec{b}\]
Special Matrices
Zero Matrix
A matrix whose entries are all zero. It serves as the additive identity.
\[O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\]For any matrix $A$ of the same size, $A + O = A$.
Identity Matrix
A square matrix with ones on the main diagonal and zeros everywhere else. It serves as the multiplicative identity.
\[I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}\]For any matrix $A$ of compatible size, $AI = IA = A$.
Square Matrix
A matrix with the same number of rows and columns, i.e., an $n \times n$ matrix. Many important concepts – such as the inverse and the determinant – are defined only for square matrices.
Diagonal Matrix
A square matrix where every off-diagonal entry is zero.
\[D = \begin{pmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{pmatrix}\]Multiplying two diagonal matrices is especially easy: just multiply the corresponding diagonal entries.
The Transpose
Definition
The transpose of a matrix $A$, written $A^T$, is the matrix obtained by swapping rows and columns. In other words, $(A^T){ij} = A{ji}$.
\[A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \implies A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}\]If $A$ is $m \times n$, then $A^T$ is $n \times m$.
Properties of the Transpose
\(\left(A^T\right)^T = A\) \((A + B)^T = A^T + B^T\) \((cA)^T = cA^T\) \((AB)^T = B^T A^T \quad \text{(note the reversal of order!)}\)
Symmetric Matrices
A square matrix that equals its own transpose is called a symmetric matrix.
\[A^T = A \quad \Leftrightarrow \quad a_{ij} = a_{ji}\]Symmetric matrices look the same on either side of the main diagonal. They appear everywhere in practice: covariance matrices in statistics, adjacency matrices of undirected graphs, and many more.
\[S = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 4 \\ 3 & 4 & 6 \end{pmatrix}\]Conversely, a matrix satisfying $A^T = -A$ is called a skew-symmetric matrix.
Key Takeaways
| Concept | Description / Formula |
|---|---|
| $m \times n$ matrix | Rectangular array with $m$ rows, $n$ columns; entries $a_{ij}$ |
| Zero matrix | All entries are 0; $A + O = A$ |
| Identity matrix | Diagonal entries are 1, rest 0; $AI = IA = A$ |
| Square matrix | Row count = column count ($n \times n$) |
| Diagonal matrix | All off-diagonal entries are 0 |
| Transpose | $(A^T){ij} = A{ji}$; $(AB)^T = B^T A^T$ |
| Symmetric matrix | $A^T = A$, i.e., $a_{ij} = a_{ji}$ |
In the next post, we explore matrix operations – addition, subtraction, scalar multiplication, and the rules of matrix multiplication.