Matrix notation
Get to grips with matrix notation for Year 11 Specialist Mathematics in Queensland (QCAA). A matrix is a rectangular array of numbers, and this is where you learn to read one: its order, its elements, and the special matrices.
You will learn to state the order of a matrix, address any element by row and column, recognise row, column, square, zero, diagonal and identity matrices, and decide when two matrices are equal — the foundation for all the matrix arithmetic and algebra that follows.
Theory
Matrix notation is the language for describing a matrix — a rectangular array of numbers — in Year 11 Specialist Mathematics (QCAA, Queensland). It covers a matrix’s order (its size), how to address an element \(a_{ij}\), the special matrices, and when two matrices are equal.
A matrix is a rectangular array of numbers set inside brackets and named with a capital letter, for example \(A=\begin{pmatrix} 6 & -2 & 3 \\ 1 & 8 & -5 \end{pmatrix}\). Each number inside it is an element (or entry).
The order (or dimensions) of a matrix is written rows \(\times\) columns, as \(m\times n\). The matrix \(A\) above has \(2\) rows and \(3\) columns, so its order is \(2\times 3\). Order always lists rows first, so a \(2\times 3\) matrix is a different size from a \(3\times 2\) matrix.
A single element is written \(a_{ij}\): the entry in row \(i\), column \(j\) — row first, then column. For \(A\), the element \(a_{23}=-5\).
Some matrices have special names: a row matrix has one row, a column matrix has one column, a square matrix has equal numbers of rows and columns, the zero matrix \(O\) has every entry \(0\), a diagonal matrix has every off-diagonal entry \(0\), and the identity matrix \(I\) has \(1\)s on the leading diagonal and \(0\)s elsewhere. Two matrices are equal only when they have the same order and equal corresponding entries.
A matrix of order \(m\times n\) has \(m\) rows and \(n\) columns; its entry in row \(i\), column \(j\) is \(a_{ij}\):
Two matrices \(A\) and \(B\) are equal exactly when they match in both order and every corresponding entry:
How to describe a matrix
- Count the rows (the horizontal lines of numbers).
- Count the columns (the vertical lines), then state the order as rows \(\times\) columns.
- Address an element \(a_{ij}\) by going down to row \(i\), then across to column \(j\) — row first, then column.
- Name it if it is special: row, column, square, zero, diagonal or identity.
Count the horizontal rows, then the vertical columns, and write order as rows \(\times\) columns:
| \(\text{rows}\) | \(=\) | \(2\) |
| \(\text{columns}\) | \(=\) | \(4\) |
| \(\text{order}\) | \(=\) | \(\text{rows}\times\text{columns}\) |
| \(=\) | \(2\times 4\) |
\(Q\) has order \(2\times 4\).
The first index is the row and the second is the column, so go to row \(3\), then column \(2\):
| \(a_{32}\) | \(=\) | \(\text{row }3,\ \text{column }2\) |
| \(\text{row }3\) | \(=\) | \((7,\ 3,\ -5)\) |
| \(\text{2nd entry}\) | \(=\) | \(3\) |
| \(a_{32}\) | \(=\) | \(3\) |
\(a_{32}=3\).
The identity has \(1\)s on the leading diagonal and \(0\)s everywhere else:
| \(\text{diagonal entries}\) | \(=\) | \(1\) |
| \(\text{off-diagonal entries}\) | \(=\) | \(0\) |
| \(I\) | \(=\) | \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) |
| \(\text{order}\) | \(=\) | \(2\times 2\) |
\(I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\), of order \(2\times 2\).
Equal matrices have the same order and equal corresponding entries, so match the entries one at a time:
| \(x\) | \(=\) | \(5\quad(\text{match } a_{12})\) |
| \(y\) | \(=\) | \(8\quad(\text{match } a_{21})\) |
\(x=5\) and \(y=8\).
Common pitfalls
Frequently asked questions
What is the order of a matrix?
The order (or dimensions) is the number of rows times the number of columns, written \(m\times n\). A matrix with \(2\) rows and \(3\) columns has order \(2\times 3\).
What does \(a_{ij}\) mean?
It is the element in row \(i\) and column \(j\) — row first, then column. For example \(a_{23}\) is the entry in row \(2\), column \(3\).
When are two matrices equal?
They must have the same order and equal corresponding entries. If the orders differ, the matrices cannot be equal.
What is a square matrix?
A square matrix has the same number of rows as columns, such as a \(2\times 2\) or \(3\times 3\) matrix.
What is the difference between a row matrix and a column matrix?
A row matrix has a single row (order \(1\times n\)); a column matrix has a single column (order \(m\times 1\)).
What is the identity matrix?
The identity matrix \(I\) is a square matrix with \(1\)s on the leading diagonal and \(0\)s elsewhere, for example \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\).