Matrices notes: ICSE Class 10 Mathematics

1. Matrices and their order

A matrix is a rectangular arrangement of numbers in rows and columns, written inside square brackets and named with a capital letter. Each number is an element; $a_{ij}$ is the element in row $i$ and column $j$ .

A matrix with $m$ rows and $n$ columns has order $m \times n$ (rows first) and $mn$ elements.
$\begin{bmatrix} 6 & -1 & 2 \end{bmatrix}$ has order $1 \times 3$ ; $\begin{bmatrix} 4 \\ 9 \end{bmatrix}$ has order $2 \times 1$ ; $\begin{bmatrix} 2 & 0 \\ -3 & 5 \end{bmatrix}$ has order $2 \times 2$ .

Example: a canteen's sales over two days, with one row per day and one column per dish, form a $2 \times 2$ matrix.

Order $=$ rows $\times$ columns, always rows first.
Matrix	Rows	Columns	Order
$\begin{bmatrix} 6 & -1 & 2 \end{bmatrix}$	1	3	$1 \times 3$
$\begin{bmatrix} 4 \\ 9 \end{bmatrix}$	2	1	$2 \times 1$
$\begin{bmatrix} 2 & 0 \\ -3 & 5 \end{bmatrix}$	2	2	$2 \times 2$
$\begin{bmatrix} 1 & 0 & 4 \\ -2 & 7 & 3 \end{bmatrix}$	2	3	$2 \times 3$

Element positions in $\begin{bmatrix} 3 & -5 \\ 0 & 8 \end{bmatrix}$: row first, then column.
	Column 1	Column 2
Row 1	$a_{11} = 3$	$a_{12} = -5$
Row 2	$a_{21} = 0$	$a_{22} = 8$

The same numbers as a matrix: $\begin{bmatrix} 30 & 18 \\ 25 & 22 \end{bmatrix}$, order $2 \times 2$.
Day	Idli	Dosa
Monday	30	18
Tuesday	25	22

Formulas

Order and size
$\text{order} = m \times n, \quad \text{number of elements} = mn$

Key terms

Matrix: A rectangular arrangement of numbers in rows and columns.
Element: One number in a matrix; $a_{ij}$ is in row $i$ , column $j$ .
Order: Rows $\times$ columns, written $m \times n$ .

2. Types of matrices

Row matrix: only one row, e.g. $\begin{bmatrix} 3 & -2 \end{bmatrix}$ .
Column matrix: only one column, e.g. $\begin{bmatrix} 5 \\ 1 \end{bmatrix}$ .
Square matrix: as many rows as columns, e.g. order $2 \times 2$ .
Zero (null) matrix $O$ : every element is $0$ (any order).
Identity matrix $I$ : a square matrix with $1$ on the leading diagonal (top left to bottom right) and $0$ elsewhere; of order $2$ it is $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ .

Every identity matrix is square, but a square matrix need not be an identity matrix.

Five types of matrix used in this chapter.
Type	What makes it	Example
Row matrix	only one row	$\begin{bmatrix} 3 & -2 \end{bmatrix}$
Column matrix	only one column	$\begin{bmatrix} 5 \\ 1 \end{bmatrix}$
Square matrix	as many rows as columns	$\begin{bmatrix} 4 & 1 \\ -6 & 2 \end{bmatrix}$
Zero (null) matrix	every element is $0$	$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
Identity matrix $I$	square; $1$ on the leading diagonal, $0$ elsewhere	$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Naming a matrix by its shape and its elements.

Formulas

Identity and zero matrices of order 2
$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Key terms

Row matrix: A matrix with only one row (order $1 \times n$ ).
Column matrix: A matrix with only one column (order $m \times 1$ ).
Square matrix: A matrix with as many rows as columns.
Zero (null) matrix: A matrix whose elements are all $0$ , written $O$ .
Identity matrix: A square matrix with $1$ on the leading diagonal and $0$ elsewhere, written $I$ .

3. Equality of matrices

Two matrices are equal when they have the same order and equal corresponding elements. Equality gives one equation for each position, which is how unknowns are found.

Example: $\begin{bmatrix} x + y & -2 \\ 6 & x - y \end{bmatrix} = \begin{bmatrix} 11 & -2 \\ 6 & 3 \end{bmatrix}$ gives $x + y = 11$ and $x - y = 3$ , so $x = 7$ and $y = 4$ .

$\begin{bmatrix} 2 & 5 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 5 \end{bmatrix}$ hold the same numbers but are not equal: their orders differ.

One equation for each position that holds an unknown.
Position	Left	Right	Equation
Row 1, column 1	$x + y$	$11$	$x + y = 11$
Row 1, column 2	$-2$	$-2$	already equal
Row 2, column 1	$6$	$6$	already equal
Row 2, column 2	$x - y$	$3$	$x - y = 3$

Equal matrices: same order and equal corresponding elements.
Pair	Equal?	Why
$\begin{bmatrix} 2 & 5 \\ -1 & 0 \end{bmatrix}$ and $\begin{bmatrix} 2 & 5 \\ -1 & 0 \end{bmatrix}$	yes	same order, same element in every position
$\begin{bmatrix} 2 & 5 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 5 \end{bmatrix}$	no	orders $1 \times 2$ and $2 \times 1$ differ
$\begin{bmatrix} 3 & 1 \\ 0 & 4 \end{bmatrix}$ and $\begin{bmatrix} 3 & 0 \\ 1 & 4 \end{bmatrix}$	no	same order, but the elements in two positions differ

The two equations from equal elements meet at $(7, 4)$.

Key terms

Equal matrices: Matrices of the same order whose corresponding elements are equal.
Corresponding elements: Elements in the same row and column of two matrices of the same order.

4. Compatibility for addition and multiplication

Addition and subtraction: $A + B$ and $A - B$ are defined only when $A$ and $B$ have the same order; the result has that order.
Multiplication: $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$ . If $A$ is $m \times n$ and $B$ is $n \times p$ , then $AB$ is $m \times p$ .

Examples: a $2 \times 2$ matrix times a $2 \times 1$ matrix gives a $2 \times 1$ matrix; a $1 \times 2$ matrix times a $2 \times 2$ matrix gives a $1 \times 2$ matrix; a $2 \times 1$ matrix times a $2 \times 2$ matrix is not defined.

Inner numbers must match; the outer numbers give the order.
First matrix	Second matrix	Product defined?	Order of product
$2 \times 2$	$2 \times 2$	yes	$2 \times 2$
$2 \times 2$	$2 \times 1$	yes	$2 \times 1$
$1 \times 2$	$2 \times 2$	yes	$1 \times 2$
$2 \times 1$	$2 \times 2$	no	not defined
$2 \times 2$	$1 \times 2$	no	not defined

Compatibility for addition and subtraction.

Compatibility for multiplication.

Formulas

Order of a product
$(m \times n)(n \times p) \to m \times p$

Key terms

Compatible for addition: Two matrices of the same order.
Compatible for multiplication: $AB$ is defined when columns of $A$ $=$ rows of $B$ .

5. Addition, subtraction and multiplication by a number

Sum and difference: add or subtract corresponding elements: $\begin{bmatrix} 4 & -3 \\ 0 & 7 \end{bmatrix} + \begin{bmatrix} -2 & 5 \\ 3 & -6 \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 3 & 1 \end{bmatrix}$ .
Scalar multiple: for a number $k$ (an integer or a non-zero rational number), $kA$ multiplies every element by $k$ : $\dfrac{1}{2}\begin{bmatrix} 8 & -2 \\ 6 & 4 \end{bmatrix} = \begin{bmatrix} 4 & -1 \\ 3 & 2 \end{bmatrix}$ .
$A + B = B + A$ , $A + O = A$ , $A + (-A) = O$ .
Matrix equations are solved like linear equations: $2A + X = B$ gives $X = B - 2A$ ; $\dfrac{1}{2}X + A = B$ gives $X = 2(B - A)$ .

Add or subtract position by position; watch the signs.
	Element by element	Result
$A + B$	$\begin{bmatrix} 4 + (-2) & -3 + 5 \\ 0 + 3 & 7 + (-6) \end{bmatrix}$	$\begin{bmatrix} 2 & 2 \\ 3 & 1 \end{bmatrix}$
$A - B$	$\begin{bmatrix} 4 - (-2) & -3 - 5 \\ 0 - 3 & 7 - (-6) \end{bmatrix}$	$\begin{bmatrix} 6 & -8 \\ -3 & 13 \end{bmatrix}$

Scalar multiples, the zero matrix and the negative of a matrix.
Rule	Example
$kA$ : multiply every element by $k$	$3\begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 0 & 12 \end{bmatrix}$
$k$ may be any non-zero rational number	$\dfrac{1}{2}\begin{bmatrix} 8 & -2 \\ 6 & 4 \end{bmatrix} = \begin{bmatrix} 4 & -1 \\ 3 & 2 \end{bmatrix}$
$A + O = A$	$\begin{bmatrix} 1 & -4 \\ 3 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & -4 \\ 3 & 2 \end{bmatrix}$
$A + (-A) = O$	$\begin{bmatrix} 1 & -4 \\ 3 & 2 \end{bmatrix} + \begin{bmatrix} -1 & 4 \\ -3 & -2 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Scalar multiples first, then subtract.
Step	Result
$3P$	$\begin{bmatrix} 6 & 0 \\ -3 & 9 \end{bmatrix}$
$\dfrac{1}{2}Q$	$\begin{bmatrix} 2 & -3 \\ 1 & 4 \end{bmatrix}$
$3P - \dfrac{1}{2}Q$	$\begin{bmatrix} 6 - 2 & 0 - (-3) \\ -3 - 1 & 9 - 4 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ -4 & 5 \end{bmatrix}$

Treat the matrix equation like a linear equation.
Step	Working
Make $X$ the subject	$X = B - 2A$
$2A$	$\begin{bmatrix} 8 & -4 \\ 2 & 6 \end{bmatrix}$
$X = B - 2A$	$\begin{bmatrix} 5 - 8 & 1 - (-4) \\ -2 - 2 & 10 - 6 \end{bmatrix} = \begin{bmatrix} -3 & 5 \\ -4 & 4 \end{bmatrix}$
Check $2A + X$	$\begin{bmatrix} 8 & -4 \\ 2 & 6 \end{bmatrix} + \begin{bmatrix} -3 & 5 \\ -4 & 4 \end{bmatrix} = \begin{bmatrix} 5 & 1 \\ -2 & 10 \end{bmatrix}$

Adding the two weekly matrices: $\begin{bmatrix} 12 & 9 \\ 7 & 15 \end{bmatrix} + \begin{bmatrix} 10 & 6 \\ 11 & 8 \end{bmatrix} = \begin{bmatrix} 22 & 15 \\ 18 & 23 \end{bmatrix}$.
Class	Fiction (week 1 + week 2)	Non-fiction (week 1 + week 2)
Class 9	$12 + 10 = 22$	$9 + 6 = 15$
Class 10	$7 + 11 = 18$	$15 + 8 = 23$

Formulas

Sum, difference, scalar multiple
$A \pm B = [a_{ij} \pm b_{ij}], \quad kA = [k a_{ij}]$

Key terms

Scalar multiple: $kA$ : every element of $A$ multiplied by the number $k$ .
Negative of a matrix: $-A$ : every element of $A$ with its sign changed; $A + (-A) = O$ .

6. Multiplying matrices

Row-by-column rule: the element in row $i$ and column $j$ of $AB$ is row $i$ of $A$ times column $j$ of $B$ (multiply in pairs and add):

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} ap + br & aq + bs \\ cp + dr & cq + ds \end{bmatrix}$ , and $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} p \\ r \end{bmatrix} = \begin{bmatrix} ap + br \\ cp + dr \end{bmatrix}$ .

Example: $\begin{bmatrix} 40 & 25 \\ 55 & 30 \end{bmatrix}\begin{bmatrix} 30 \\ 50 \end{bmatrix} = \begin{bmatrix} 2450 \\ 3150 \end{bmatrix}$ gives a stall's takings for two days.

Unknowns from a product: multiply out and equate: $\begin{bmatrix} 3 & 1 \\ 1 & -2 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 11 \\ -1 \end{bmatrix}$ gives $3x + y = 11$ and $x - 2y = -1$ , so $x = 3$ , $y = 2$ .

Each element of $AB$: a row of $A$ times a column of $B$.
Element of $AB$	Row of $A$	Column of $B$	Value
Row 1, column 1	$a, b$	$p, r$	$ap + br$
Row 1, column 2	$a, b$	$q, s$	$aq + bs$
Row 2, column 1	$c, d$	$p, r$	$cp + dr$
Row 2, column 2	$c, d$	$q, s$	$cq + ds$

Each row of $SP$ is one day's takings.
Day	Row of $S$ times $P$	Takings (₹)
Saturday	$40 \times 30 + 25 \times 50 = 2450$	2450
Sunday	$55 \times 30 + 30 \times 50 = 3150$	3150

Same two matrices, different products.
Product	Row-by-column working	Result
$AB$	$\begin{bmatrix} 3 \times 0 + (-1) \times 1 & 3 \times 2 + (-1) \times (-3) \\ 2 \times 0 + 4 \times 1 & 2 \times 2 + 4 \times (-3) \end{bmatrix}$	$\begin{bmatrix} -1 & 9 \\ 4 & -8 \end{bmatrix}$
$BA$	$\begin{bmatrix} 0 \times 3 + 2 \times 2 & 0 \times (-1) + 2 \times 4 \\ 1 \times 3 + (-3) \times 2 & 1 \times (-1) + (-3) \times 4 \end{bmatrix}$	$\begin{bmatrix} 4 & 8 \\ -3 & -13 \end{bmatrix}$

The two equations from the product meet at $(3, 2)$.

Formulas

Element of a product (order 2)
$(AB)_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j}$

Key terms

Row-by-column rule: Each element of $AB$ is a row of $A$ times a column of $B$ .

7. Properties of matrix multiplication

Not commutative: in general $AB \ne BA$ (and one of them may not even be defined).
Identity: $AI = IA = A$ for every square matrix $A$ of order $2$ .
Zero products: $AB = O$ is possible with $A \ne O$ and $B \ne O$ .
Squares: $A^2 = A \times A$ (row by column, not the square of each element), and $(A + B)^2 = A^2 + AB + BA + B^2$ .

Example: for $A = \begin{bmatrix} 2 & 3 \\ -1 & 1 \end{bmatrix}$ , $A^2 = \begin{bmatrix} 1 & 9 \\ -3 & -2 \end{bmatrix}$ and $A^2 - 3A + 5I = O$ .

Not in the ICSE 2027 syllabus: adding, subtracting or multiplying matrices of order larger than $2 \times 2$ ; in this chapter all such working stays within order $2 \times 2$ .

Rules from algebra that change for matrices.
For numbers	For square matrices of order 2
$ab = ba$	$AB \ne BA$ in general
$ab = 0$ means $a = 0$ or $b = 0$	$AB = O$ is possible with $A \ne O$ and $B \ne O$
$(a + b)^2 = a^2 + 2ab + b^2$	$(A + B)^2 = A^2 + AB + BA + B^2$
$1 \times a = a \times 1 = a$	$IA = AI = A$

$AI = IA = A$: the identity matrix acts like the number $1$.
Product	Working	Result
$AI$	$\begin{bmatrix} 5 \times 1 + (-2) \times 0 & 5 \times 0 + (-2) \times 1 \\ 3 \times 1 + 7 \times 0 & 3 \times 0 + 7 \times 1 \end{bmatrix}$	$\begin{bmatrix} 5 & -2 \\ 3 & 7 \end{bmatrix}$
$IA$	$\begin{bmatrix} 1 \times 5 + 0 \times 3 & 1 \times (-2) + 0 \times 7 \\ 0 \times 5 + 1 \times 3 & 0 \times (-2) + 1 \times 7 \end{bmatrix}$	$\begin{bmatrix} 5 & -2 \\ 3 & 7 \end{bmatrix}$

Two non-zero matrices with $AB = O$.
Product	Working	Result
$AB$	$\begin{bmatrix} 1 \times 3 + (-1) \times 3 & 1 \times 1 + (-1) \times 1 \\ (-2) \times 3 + 2 \times 3 & (-2) \times 1 + 2 \times 1 \end{bmatrix}$	$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
$BA$	$\begin{bmatrix} 3 \times 1 + 1 \times (-2) & 3 \times (-1) + 1 \times 2 \\ 3 \times 1 + 1 \times (-2) & 3 \times (-1) + 1 \times 2 \end{bmatrix}$	$\begin{bmatrix} 1 & -1 \\ 1 & -1 \end{bmatrix}$

Square first, then combine element by element.
Step	Result
$A^2$	$\begin{bmatrix} 2 \times 2 + 3 \times (-1) & 2 \times 3 + 3 \times 1 \\ (-1) \times 2 + 1 \times (-1) & (-1) \times 3 + 1 \times 1 \end{bmatrix} = \begin{bmatrix} 1 & 9 \\ -3 & -2 \end{bmatrix}$
$3A$	$\begin{bmatrix} 6 & 9 \\ -3 & 3 \end{bmatrix}$
$5I$	$\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$
$A^2 - 3A + 5I$	$\begin{bmatrix} 1 - 6 + 5 & 9 - 9 + 0 \\ -3 - (-3) + 0 & -2 - 3 + 5 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

In general $AB \ne BA$: the order of the factors matters.
Position	$AB$	$BA$
Row 1, column 1	$-1$	$4$
Row 1, column 2	$9$	$8$
Row 2, column 1	$4$	$-3$
Row 2, column 2	$-8$	$-13$

Formulas

Identity
$AI = IA = A$
Square of a sum
$(A + B)^2 = A^2 + AB + BA + B^2$

Key terms

Non-commutative: $AB$ and $BA$ are in general different.
$A^2$: The product $A \times A$ of a square matrix with itself.

Exam tips

How marks are awarded

Order, type and equality (1–2 marks): order rows by columns 1; each unknown 1.
Sums, differences and matrix equations (2–3 marks): rearranged 1; scalar multiple 1; answer 1.
Products (2–4 marks): orders checked and working shown 1; each correct row or element 1.
"Show that" with $A^2$ (3–4 marks): $A^2$ 1–2; other terms 1; conclusion 1.
10-mark structured question: parts (a), (b), (c) usually 3 + 3 + 4; show all working.

Common mistakes

Writing the order as columns by rows.
Adding matrices of different orders, or multiplying when columns of the first $\ne$ rows of the second.
Multiplying corresponding elements instead of rows by columns.
Squaring each element to find $A^2$ .
Assuming $AB = BA$ , or that $AB = O$ forces $A = O$ or $B = O$ .
Sign slips when subtracting negative elements.

1. Matrices and their order

Formulas

Key terms

2. Types of matrices

Formulas

Key terms

3. Equality of matrices

Key terms

4. Compatibility for addition and multiplication

Formulas

Key terms

5. Addition, subtraction and multiplication by a number

Formulas

Key terms

6. Multiplying matrices

Formulas

Key terms

7. Properties of matrix multiplication

Formulas

Key terms

Exam tips

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