Matrices • Topic 1 of 3

Matrices: Order, Types & Equality

A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with $m$ rows and $n$ columns is said to have order $m\times n$, and the entry in row $i$, column $j$ is written $a_{ij}$. Matrices give us a compact language for systems of equations, transformations and data tables.

Special types

Type	Defining condition
Row matrix	only one row ($1\times n$)
Column matrix	only one column ($m\times 1$)
Square matrix	$m=n$
Diagonal matrix	square, all off-diagonal entries $0$
Scalar matrix	diagonal with all diagonal entries equal
Identity $I_n$	scalar matrix with diagonal entries $1$
Zero matrix $O$	every entry $0$

Equality of matrices

Two matrices are equal only if they have the same order and every corresponding entry is equal: $A=B \iff a_{ij}=b_{ij}$ for all $i,j$. This single idea turns one matrix equation into a set of scalar equations — the standard way "find $x,y$" problems are solved.

How many entries?

An $m\times n$ matrix has $mn$ entries. If you must build a matrix with exactly $12$ entries, the possible orders are the factor pairs of $12$: $1\times12,\ 2\times6,\ 3\times4,\ 4\times3,\ 6\times2,\ 12\times1$ — six in all.

Solved Examples

Worked Example

If $\begin{bmatrix} x+y & 2\\ 5 & x-y\end{bmatrix}=\begin{bmatrix} 7 & 2\\ 5 & 3\end{bmatrix}$, find $x$ and $y$.

Solution

Equality entry-by-entry gives $x+y=7$ and $x-y=3$. Adding: $2x=10\Rightarrow x=5$; then $y=2$.

Worked Example

Construct the $2\times2$ matrix $A=[a_{ij}]$ where $a_{ij}=\dfrac{(i+j)^2}{2}$.

Solution

$a_{11}=\tfrac{(2)^2}{2}=2,\ a_{12}=\tfrac{(3)^2}{2}=\tfrac92,\ a_{21}=\tfrac{9}{2},\ a_{22}=\tfrac{(4)^2}{2}=8$. So $A=\begin{bmatrix} 2 & \tfrac92\\[2pt] \tfrac92 & 8\end{bmatrix}$.

Worked Example

How many matrices of order $2\times3$ can be formed with each entry $0$ or $1$?

Solution

There are $2\times3=6$ positions and each independently has $2$ choices, giving $2^{6}=64$ matrices.

Worked Example

Write the orders possible for a matrix having exactly $7$ elements.

Solution

$7$ is prime, so the only factor pairs are $1\times7$ and $7\times1$ — just two possible orders.

Key Points

Order $m\times n$ means $m$ rows, $n$ columns; entry $a_{ij}$ sits in row $i$, column $j$.
Diagonal $\subset$ scalar $\subset$ identity is the nesting for square matrices with the obvious extra conditions.
$A=B$ requires equal order and equal corresponding entries.
An $m\times n$ matrix has $mn$ entries; possible orders for $N$ entries = factor pairs of $N$.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.A matrix has $18$ elements. How many distinct orders are possible?

Explanation: Factor pairs of $18$: $1\times18,2\times9,3\times6,6\times3,9\times2,18\times1$ — six orders.

Q2.A scalar matrix is always a:

Explanation: A scalar matrix is diagonal with equal diagonal entries, so it is a special diagonal matrix.

Q3.If $\begin{bmatrix}a & 3\\ 0 & b\end{bmatrix}=\begin{bmatrix}2 & 3\\ 0 & 5\end{bmatrix}$, then $a+b=$

Explanation: $a=2,\ b=5\Rightarrow a+b=7$.

Q4.The number of $2\times2$ matrices with entries from $\{0,1\}$ is:

Explanation: $4$ positions, $2$ choices each: $2^4=16$.

Practice more MCQs → 📝 Assignment · 35 marks