Matrix definition and types

A matrix is a rectangular arrangement of numbers, symbols, or expressions, arranged in rows and columns. The numbers are called entries of the matrix. We use capital letters $A, B, C, \ldots$ for matrices and lowercase indexed letters $a_{ij}$ for the entry in row $i$, column $j$. The matrix $A$ with $m$ rows and $n$ columns has order (or dimension) $m \times n$.

Formal definition

$A = (a_{ij})_{m \times n} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}.$

When we say $A$ has order $m \times n$, $m$ comes first. The number of entries is $mn$. Two matrices are equal if they have the same order and equal entries in every position.

Named special matrices

Examples

A $2 \times 3$ matrix: $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$.

A row matrix of order $1 \times 4$: $(7 \quad 8 \quad 9 \quad 10)$.

A diagonal matrix of order $3 \times 3$: $\begin{pmatrix} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{pmatrix}$.

The $3 \times 3$ identity: $I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$.

Constructing matrices from formulas

Often a matrix is defined by giving a formula for $a_{ij}$. For example, $A = (a_{ij})_{2 \times 3}$ with $a_{ij} = i + 2j$ gives:

$a_{11} = 3, a_{12} = 5, a_{13} = 7, a_{21} = 4, a_{22} = 6, a_{23} = 8$.

So $A = \begin{pmatrix} 3 & 5 & 7 \\ 4 & 6 & 8 \end{pmatrix}$.

Equality of matrices

Two matrices $A = (a_{ij})$ and $B = (b_{ij})$ are equal, written $A = B$, if and only if:

1. They have the same order, and
2. $a_{ij} = b_{ij}$ for all $i, j$.

So $\begin{pmatrix} 1 & 2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ are not equal , different orders.

Counting matrices

The number of distinct $m \times n$ matrices with entries from a set $S$ of size $k$ is $k^{mn}$. For example, $2 \times 2$ binary matrices number $2^4 = 16$.

Worked examples

Example 1. Find the order of $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}$. State the entry $a_{32}$.

Order $3 \times 2$. $a_{32} = 6$.

Example 2. Construct $B = (b_{ij})_{2 \times 2}$ with $b_{ij} = (i - j)^2$.

$b_{11} = 0, b_{12} = 1, b_{21} = 1, b_{22} = 0$. So $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

Example 3. If $A = \begin{pmatrix} 2 & 3 \\ x & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & y \\ 4 & 5 \end{pmatrix}$ are equal, find $x$ and $y$.

$x = 4, y = 3$.

Example 4. Identify the order of the zero matrix needed to add to $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ such that the result is itself.

The matrix is $2 \times 2$, so the zero matrix of order $2 \times 2$ adds with it.

Example 5. Find a $3 \times 3$ diagonal matrix with diagonal entries $1, 2, 3$. Is it also a scalar matrix?

$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$. Not a scalar matrix (diagonal entries differ).

Example 6. Write down a $3 \times 3$ upper triangular matrix all of whose diagonal entries are zero.

$\begin{pmatrix} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{pmatrix}$ for any $a, b, c$.

Try it yourself

1. State the order of $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$.
2. Construct the $3 \times 3$ matrix $A = (a_{ij})$ with $a_{ij} = i + j$.
3. State the entry $a_{23}$ of the matrix in the previous question.
4. If $A = \begin{pmatrix} x & 2 \\ 3 & y \end{pmatrix} = \begin{pmatrix} 5 & 2 \\ 3 & 7 \end{pmatrix}$, find $x, y$.
5. Write the $2 \times 2$ identity matrix.
6. Find the number of $2 \times 2$ matrices with entries from $\{0, 1, 2\}$.
7. Construct a $4 \times 4$ scalar matrix with scalar $-3$.
8. Is the matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ upper or lower triangular? Neither?
9. Construct a $3 \times 3$ lower triangular matrix where all diagonal entries are $1$.
10. If a matrix has order $5 \times 7$, how many entries does it have?
11. Write down a $2 \times 2$ matrix that is both upper and lower triangular.
12. State whether $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is a scalar matrix. Justify.
13. Construct a $3 \times 3$ matrix $A$ with $a_{ij} = i \cdot j$.
14. List all possible orders for a matrix with exactly $12$ entries.

Pitfalls / Tricks

• The number of entries is $m \cdot n$, but the order is always written $m \times n$ (rows first).
• A matrix that is both upper and lower triangular must be diagonal.
• The zero matrix and the identity matrix are not the same , be careful in operations.
• Each entry $a_{ij}$ is identified by row $i$, column $j$. Confusing these is a frequent mistake.
• A scalar matrix is a special case of a diagonal matrix, but not every diagonal matrix is scalar.

Next we define addition and multiplication on matrices.