A Matrix Is a Neat Table of Numbers That Follows Its Own Arithmetic

A matrix is a rectangular arrangement of numbers in rows and columns, written inside brackets and named with a capital letter:

$$A=\left[\begin{array}{ccc}2& -1& 5\\ 0& 3& 4\end{array}\right]$$

Each number is an element. Arranging data this way lets you add, compare and scale whole tables at once, following clear rules.

This part covers order and types, equality, addition and subtraction, and multiplication by a number.

The order of a matrix is written as rows × columns; its type depends on its shape and its elements.

Order. The matrix $A$ above has $2$ rows and $3$ columns, so its order is $2\times 3$ and it has $2\times 3=6$ elements.

Types of matrices:

- Row matrix — only one row, such as $\left[\begin{array}{cc}4& 7\end{array}\right]$, order $1\times 2$
- Column matrix — only one column, such as $\left[\begin{array}{c}4\\ 7\end{array}\right]$, order $2\times 1$
- Square matrix — equal numbers of rows and columns, such as order $2\times 2$
- Zero (null) matrix $O$ — every element is $0$
- Identity matrix $I$ — a square matrix with $1$ on the leading diagonal and $0$ elsewhere: $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Worked example. A matrix has $6$ elements. What orders are possible?

$$1\times 6,6\times 1,2\times 3,3\times 2$$

An everyday example. **A class teacher records the marks of $3$ students in Maths and Science** as a table with $3$ rows and $2$ columns — a $3\times 2$ matrix.

The substance. Rows always come first: a $2\times 3$ matrix and a $3\times 2$ matrix are different shapes.

Two matrices are equal only if they have the same order and every pair of corresponding elements is equal, so equate matching elements and solve.

Worked example 1.

$$\left[\begin{array}{cc}x+2& 5\\ 3& y-1\end{array}\right]=\left[\begin{array}{cc}7& 5\\ 3& 4\end{array}\right]x+2=7,y-1=4x=5,y=5$$

Worked example 2.

$$\left[\begin{array}{cc}2a+b& 3\\ a-b& 4\end{array}\right]=\left[\begin{array}{cc}7& 3\\ 2& 4\end{array}\right]$$

$$2a+b=7\text{and}a-b=2\text{adding: }3a=9a=3,b=1$$

An everyday example. Two copies of a school canteen price list are the same only if every item has the same price in the same position.

The boundary case. **$\left[\begin{array}{cc}1& 2\end{array}\right]$ and $\left[\begin{array}{c}1\\ 2\end{array}\right]$ are not equal**, even though they contain the same numbers, because their orders differ.

Two matrices can be added or subtracted only if they have the same order; then you add or subtract the corresponding elements.

Worked example. Let

$$A=\left[\begin{array}{cc}3& -2\\ 1& 4\end{array}\right],B=\left[\begin{array}{cc}-1& 5\\ 2& 0\end{array}\right]$$

$$A+B=\left[\begin{array}{cc}3-1& -2+5\\ 1+2& 4+0\end{array}\right]=\left[\begin{array}{cc}2& 3\\ 3& 4\end{array}\right]$$

$$A-B=\left[\begin{array}{cc}3+1& -2-5\\ 1-2& 4-0\end{array}\right]=\left[\begin{array}{cc}4& -7\\ -1& 4\end{array}\right]$$

Useful facts:

- **$A+B=B+A$ — addition is commutative
- $A+O=A$ — adding the zero matrix changes nothing
- A $2\times 2$ matrix cannot be added to a $2\times 1$ matrix

An everyday example. A stationery shop records pens and notebooks sold by two branches in two weeks. Adding the week 1 matrix to the week 2 matrix gives the fortnight's sales in one step.

The substance. Subtraction is not commutative**: $B-A$ has every element of $A-B$ with its sign changed.

Multiplying a matrix by a number multiplies every element by that number; to solve a matrix equation, rearrange it like an ordinary equation and then apply scalar multiplication and equality.

Worked example 1. With $A$ and $B$ above:

$$3A=\left[\begin{array}{cc}9& -6\\ 3& 12\end{array}\right]\frac{1}{2}B=\left[\begin{array}{cc}-\frac{1}{2}& \frac{5}{2}\\ 1& 0\end{array}\right]$$

Worked example 2. Find $X$ if $2X+A=B$.

$$2X=B-A=\left[\begin{array}{cc}-4& 7\\ 1& -4\end{array}\right]X=\frac{1}{2}\left[\begin{array}{cc}-4& 7\\ 1& -4\end{array}\right]=\left[\begin{array}{cc}-2& \frac{7}{2}\\ \frac{1}{2}& -2\end{array}\right]$$

Worked example 3. Find $x$ and $y$.

$$2\left[\begin{array}{cc}x& 3\\ 1& y\end{array}\right]+\left[\begin{array}{cc}1& -2\\ 0& 5\end{array}\right]=\left[\begin{array}{cc}7& 4\\ 2& 11\end{array}\right]$$

$$2x+1=7\Rightarrow x=32y+5=11\Rightarrow y=3$$

The other elements check: $6-2=4$ and $2+0=2$.

An everyday example. **If every price on a shop's rate card rises by $10\%$**, multiplying the price matrix by $1.1$ gives the new card at once.

The substance. Always check the elements you did not use — they confirm the equation is consistent.