AcademyMatrix Algebra
Academy
Matrix Notation
Level 1 - Linear Algebra topic page in Matrix Algebra.
Principle
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices package many numbers into one object, so they can represent linear systems, data tables, and linear transformations.
The size of a matrix is its number of rows followed by its number of columns.
Notation
\(A\)
a matrix
\(a_{ij}\)
the entry of A in row i and column j
\(M_{m,n}(\mathbb R)\)
the set of all m by n matrices with real entries
\(m\)
the number of rows
\(n\)
the number of columns
\(\mathbf r_i\)
row i of a matrix
\(\mathbf c_j\)
column j of a matrix
The Core Method
Write an \(m\) by \(n\) matrix as
Matrix entries
\[A=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{m1}&a_{m2}&\cdots&a_{mn}\end{pmatrix}\]
The first index tells the row. The second index tells the column. Therefore \(a_{23}\) means row \(2\), column \(3\), not row \(3\), column \(2\).
Two matrices are equal exactly when they have the same size and the same matching entries:
Matrix equality
\[A=B\Longleftrightarrow a_{ij}=b_{ij}\text{ for every valid }i,j\]
Worked Cases
Question
For \(A=\begin{pmatrix}2&-1&0\\4&5&7\end{pmatrix}\), find \(a_{12}\), \(a_{21}\), and the size of \(A\).
Answer
The matrix has \(2\) rows and \(3\) columns, so \(A\in M_{2,3}(\mathbb R)\). The entry \(a_{12}\) is row \(1\), column \(2\), so \(a_{12}=-1\). The entry \(a_{21}\) is row \(2\), column \(1\), so \(a_{21}=4\).
Examples
Question
What is the matrix size of \(\begin{pmatrix}1\\0\\-3\end{pmatrix}\)?
Answer
This object has \(3\) rows and \(1\) column. It is a \(3\) by \(1\) matrix, so it lies in \(M_{3,1}(\mathbb R)\). It can also represent a vector in \(\mathbb R^3\).