Number of entries
\[\text{order } m\times n \;\Rightarrow\; mn \text{ entries}\]
Possible orders of a matrix with \(N\) entries correspond to factor pairs of \(N\). E.g. 12 entries: \(1\times12, 2\times6, 3\times4, 4\times3, 6\times2, 12\times1\).
Matrices
Matrices and Their Types
Understand
A matrix is an ordered rectangular array of numbers (or functions) called its elements or entries. A matrix with \(m\) rows and \(n\) columns has order \(m\times n\) and contains \(mn\) entries. We write \(A = [a_{ij}]_{m\times n}\), where \(a_{ij}\) is the entry in row \(i\) and column \(j\).
For example, \(A = \begin{bmatrix} 2 & -1 & 0 \\ 4 & 3 & 7 \end{bmatrix}\) has order \(2\times3\), and \(a_{21} = 4\).
Types of matrices
| Type | Description | Example |
|---|---|---|
| Row / column | Only one row / one column | \(\begin{bmatrix}1 & 5 & 2\end{bmatrix}\) |
| Square | Same number of rows and columns | \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) |
| Diagonal | Square; all off-diagonal entries 0 | \(\begin{bmatrix} 2 & 0 \\ 0 & -5 \end{bmatrix}\) |
| Scalar | Diagonal with equal diagonal entries | \(\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\) |
| Identity \(I_n\) | Scalar matrix with 1s on the diagonal | \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) |
| Zero \(O\) | All entries 0 | \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\) |
Equality of matrices
Two matrices are equal if they have the same order and every pair of corresponding entries is equal. Equality of two \(2\times2\) matrices therefore gives four equations.
Key Concepts
- Order is always written (rows) × (columns).
- A matrix with \(N\) entries can have any order \(m\times n\) with \(mn = N\).
- Identity ⊂ scalar ⊂ diagonal ⊂ square.
Formula Bank
Key Points
Order is rows × columns
Always state order as (rows) × (columns). A \(2\times3\) and a \(3\times2\) matrix are different.
Practice & Topic Test
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Order \(m\times n\) ⇒ \(mn\) entries. \(a_{ij}\): row \(i\), column \(j\). Know the six types. Equal matrices ⇒ same order and equal corresponding entries.
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