Year 11 General Matrices And Matrix Arithmetic

The Basics Of A Matrix

5 practice questions 2 video lessons Theory + worked examples

Theory

A matrix is a rectangular arrangement of numbers in rows and columns. Its order is written as rows × columns, and each element $A_{i j}$ is identified by its row $i$ and column $j$ — always row first. Key types include row, column, square, zero, and identity matrices. Two matrices are equal only if they have the same order AND identical entries.

A matrix is a rectangular arrangement of numbers in rows and columns, written between square brackets:

$A = [\begin{matrix} 2 & - 5 & 8 \\ 3 & 7 & - 1 \\ - 4 & 0 & 6 \end{matrix}]$

Each number inside is called an element (or entry).

The order (or size) of a matrix is written as "rows $\times$ columns". Matrix $A$ above has $3$ rows and $3$ columns, so its order is $3 \times 3$ . A matrix with $m$ rows and $n$ columns has order $m \times n$ and contains $m \times n$ elements in total.

The element in row $i$ , column $j$ is denoted $A_{i j}$ . Always row first, column second. For example, $A_{12} = - 5$ , $A_{23} = - 1$ , and $A_{31} = - 4$ .

Two matrices are equal only if (i) they have the same order, and (ii) every corresponding element is equal.

The first diagram is an anatomy view of a matrix — pointing to a row, a column, an element, and the order. The second shows the five special types you should be able to recognise.

3 \times 3

matrix with a row, column, and element

A_{23} = - 1

highlighted.

Row, column, square, zero, and identity matrices — the standard named types.

There are no formulas to memorise — just conventions and notation.

Order of a matrix

$order = (rows) \times (columns)$

A matrix with $m$ rows and $n$ columns is an $m \times n$ matrix and contains $m \times n$ elements in total.

order = m \times n

Subscript notation for an element

$A_{i j} = the element in row i, column j$

So $A_{12}$ is row 1, column 2 — not column 1, row 2.

A_{i j}

The $n \times n$ identity matrix

$I_{3} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Equality rule. Two matrices are equal only if they have the same order AND every corresponding element matches. A 2×3 matrix can never equal a 3×2 matrix, even if they contain the same six numbers.

How to read a matrix question

Count the rows and columns to find the order $m \times n$ .
Multiply rows by columns to get the total number of elements.
For an entry like $A_{i j}$ , locate row $i$ first, then column $j$ , and read off the value at that intersection.

Building a matrix from a rule

If asked for a matrix where $M_{i j} =$ some formula in $i$ and $j$ , compute each entry by substituting the row number and column number into the formula.
Arrange the results in the grid with row $1$ on top, column $1$ on the left.

Solving matrix-equality problems

Confirm both matrices have the same order. If they don't, they can never be equal.
Match entries one-to-one. Each pair of corresponding entries gives you one equation.
Solve those simple equations to find any unknowns.

EXAMPLE 1 — ORDER AND TOTAL ELEMENTS

What is the order of the matrix

M

below? How many elements does it contain?

M = [\begin{matrix} 3 & 7 & - 2 \\ 5 & 0 & 1 \\ 4 & 8 & 6 \\ - 1 & 2 & 9 \end{matrix}]

SOLUTION

Count the rows and columns. $M$ has $4$ rows and $3$ columns, so its order is $4 \times 3$ .

Total elements	$=$	$4 \times 3$
	$=$	$12$

Answer: $M$ is a $4 \times 3$ matrix with $12$ elements.

order = 4 \times 3, elements = 12

EXAMPLE 2 — IDENTIFY AN ELEMENT

For the matrix

A = [\begin{matrix} 2 & - 5 & 8 \\ 3 & 7 & - 1 \\ - 4 & 0 & 6 \end{matrix}]

, find

A_{23}

and

A_{31}

SOLUTION

Remember: row first, column second.

$A_{23}$	$=$	row 2, column 3 $= - 1$
$A_{31}$	$=$	row 3, column 1 $= - 4$

Answer: $A_{23} = - 1$ and $A_{31} = - 4$ .

A_{23} = - 1, A_{31} = - 4

EXAMPLE 3 — BUILD FROM A RULE

Construct the

2 \times 2

matrix

M

where

M_{i j} = 2 i - j

SOLUTION

Compute each of the four entries by substituting the row and column numbers.

$M_{11}$	$=$	$2 (1) - 1 = 1$
$M_{12}$	$=$	$2 (1) - 2 = 0$
$M_{21}$	$=$	$2 (2) - 1 = 3$
$M_{22}$	$=$	$2 (2) - 2 = 2$

Arrange the entries with row 1 on top:

$M = [\begin{matrix} 1 & 0 \\ 3 & 2 \end{matrix}]$

M = [\begin{matrix} 1 & 0 \\ 3 & 2 \end{matrix}]

EXAMPLE 4 — EQUAL MATRICES

Find

x

given

[\begin{matrix} 3 & x \\ 5 & 7 \end{matrix}] = [\begin{matrix} 3 & - 2 \\ 5 & 7 \end{matrix}]

SOLUTION

Both matrices are $2 \times 2$ , so equality is possible. Match corresponding entries — the only entry that differs is row 1, column 2.

x

=

- 2

Answer: $x = - 2$ .

x = - 2

Common pitfalls

Order is rows first, then columns — always. A

2 \times 3

matrix has 2 rows and 3 columns, not the other way round. Get this backwards and everything that follows will be wrong.

The subscript $A_{i j}$ is also row first.

A_{23}

is the entry in row 2, column 3 — not column 2, row 3. A very common slip.

Same total of entries does not mean equal. A

2 \times 3

matrix and a

3 \times 2

matrix both have 6 entries, but they are different shapes and can never be equal. Orders must match exactly.

The identity matrix must be square. A non-square matrix with ones along the diagonal is not an identity matrix.

I_{n}

only exists for

n \times n

matrices.

Pay attention to brackets. Matrices are written between square brackets

[]

. Some books use round brackets

()

— both are fine, but braces

{}

are usually reserved for sets.

Frequently asked questions

What is a matrix?

A matrix is a rectangular arrangement of numbers in rows and columns, written between square brackets. Each number is called an element or entry. Matrices are used to organise data, solve systems of equations, and represent transformations.

What is the order of a matrix?

The order of a matrix is the number of rows times the number of columns, written as 'rows by columns'. A matrix with 2 rows and 3 columns has order 2 by 3. The total number of elements equals the product of the order, so a 2 by 3 matrix has 6 elements.

Which comes first in matrix notation — row or column?

Always row first, then column. The order m by n means m rows and n columns. The subscript A subscript ij means the element in row i and column j. This row-first convention applies in both contexts.

What does A subscript ij mean?

A subscript ij refers to the entry in row i, column j of matrix A. For example, A subscript 23 is the element in the second row and third column. Always read the first subscript as the row and the second as the column.

What is an identity matrix?

An identity matrix is a square matrix with ones along the main diagonal (top-left to bottom-right) and zeros everywhere else. It is denoted I or sometimes I subscript n where n is the order. The identity matrix plays the same role for matrix multiplication that the number 1 plays for ordinary multiplication.

When are two matrices equal?

Two matrices are equal only if they have the same order AND every corresponding element is equal. A 2 by 3 matrix and a 3 by 2 matrix can never be equal even if they contain the same numbers, because their orders differ.

Video Lessons

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Using Matrices To Model (Represent) Practical Situations

The Basics Of A Matrix

📖 Theory

Order of a matrix

Subscript notation for an element

The n×n identity matrix

How to read a matrix question

Building a matrix from a rule

Solving matrix-equality problems

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

The $n \times n$ identity matrix

Video Lessons

Practice Questions