Matrices notes: ISC Class 12 Mathematics

1. Order, entries and types

A matrix with $m$ rows and $n$ columns has order $m \times n$ ; $a_{ij}$ is the entry in row $i$ , column $j$ . A matrix with $mn$ entries can be arranged in as many orders as $mn$ has divisors.

Row / column matrix: one row / one column. Zero matrix $O$ : every entry $0$ .
Square ( $m = n$ ): diagonal (off-diagonal entries $0$ ), scalar (diagonal, all diagonal entries equal, $kI$ ), identity $I_n$ (scalar with $k = 1$ ), upper / lower triangular (zero below / above the diagonal).

ISC 2027 uses matrices of order at most $3$ .

Every example is checked against its condition by the build.
Type	Condition	Example
Row	one row	$\begin{bmatrix} 4 & -1 & 7 \end{bmatrix}$
Column	one column	$\begin{bmatrix} 2 \\ 9 \end{bmatrix}$
Zero	every entry $0$	$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
Diagonal	square; $a_{ij} = 0$ for $i \ne j$	$\begin{bmatrix} 6 & 0 \\ 0 & -2 \end{bmatrix}$
Scalar	diagonal, equal diagonal entries	$\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$
Identity	scalar with diagonal entries $1$	$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
Upper triangular	square; $a_{ij} = 0$ for $i > j$	$\begin{bmatrix} 1 & 5 & 2 \\ 0 & 3 & -4 \\ 0 & 0 & 6 \end{bmatrix}$
Lower triangular	square; $a_{ij} = 0$ for $i < j$	$\begin{bmatrix} 2 & 0 \\ 7 & 1 \end{bmatrix}$

Choose the most specific name the question asks for.
If a matrix is…	then it is also…
identity	scalar and diagonal
scalar	diagonal
diagonal	upper and lower triangular
square zero	diagonal, scalar and both triangular

Formulas

Notation
$A = [a_{ij}]_{m \times n}$

Key terms

Order: Rows $\times$ columns.
Scalar matrix: A diagonal matrix $kI$ with all diagonal entries equal.

2. Equality and transpose

$A = B$ when they have the same order and $a_{ij} = b_{ij}$ for every $i, j$ : one matrix equation gives several ordinary equations.

The transpose $A'$ (order $n \times m$ ) has $(A')_{ij} = a_{ji}$ .

$(A')' = A$ , $\;(A + B)' = A' + B'$ , $\;(kA)' = kA'$
$(AB)' = B'A'$ — the order reverses.

Each law is checked by the build on sample matrices.
Law	In words
$(A')' = A$	transposing twice changes nothing
$(A + B)' = A' + B'$	transpose each term
$(kA)' = kA'$	scalars pass through
$(AB)' = B'A'$	reverse the order

Row $i$ of $A$ becomes column $i$ of $A'$.
$A$	$A'$	Orders
$\begin{bmatrix} 1 & 4 & -3 \\ 2 & 0 & 5 \end{bmatrix}$	$\begin{bmatrix} 1 & 2 \\ 4 & 0 \\ -3 & 5 \end{bmatrix}$	$2 \times 3 \;\to\; 3 \times 2$

Formulas

Reversal law
$(AB)' = B'A'$

Key terms

Transpose: Rows of $A$ written as columns, written $A'$ .

3. Symmetric and skew-symmetric matrices

Symmetric: $A' = A$ ( $a_{ij} = a_{ji}$ ).
Skew-symmetric: $A' = -A$ ( $a_{ij} = -a_{ji}$ ; every diagonal entry is $0$ ).

For any square $A$ : $A + A'$ is symmetric and $A - A'$ is skew-symmetric, since $(A + A')' = A' + A$ and $(A - A')' = -(A - A')$ . Hence

A = \tfrac12 (A + A') + \tfrac12 (A - A').

This split is unique. The only matrix that is both symmetric and skew-symmetric is $O$ .

Look across the leading diagonal.
Type	Condition	Example
Symmetric	$A' = A$ , so $a_{ij} = a_{ji}$	$\begin{bmatrix} 1 & 4 & -2 \\ 4 & 0 & 7 \\ -2 & 7 & 3 \end{bmatrix}$
Skew-symmetric	$A' = -A$ , so $a_{ij} = -a_{ji}$ and $a_{ii} = 0$	$\begin{bmatrix} 0 & 5 & -1 \\ -5 & 0 & 2 \\ 1 & -2 & 0 \end{bmatrix}$

$P + Q = A$, with $P$ symmetric and $Q$ skew-symmetric.
Step	Matrix
$A'$	$\begin{bmatrix} 5 & -1 \\ 3 & 4 \end{bmatrix}$
$A + A'$	$\begin{bmatrix} 10 & 2 \\ 2 & 8 \end{bmatrix}$
$P = \tfrac12 (A + A')$	$\begin{bmatrix} 5 & 1 \\ 1 & 4 \end{bmatrix}$
$A - A'$	$\begin{bmatrix} 0 & 4 \\ -4 & 0 \end{bmatrix}$
$Q = \tfrac12 (A - A')$	$\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$

Formulas

Unique split
$A = \tfrac12 (A + A') + \tfrac12 (A - A')$

Key terms

Symmetric matrix: $A' = A$ .
Skew-symmetric matrix: $A' = -A$ ; zero diagonal.

4. Addition, scalar multiplication and multiplication

$A \pm B$ (same order): add or subtract corresponding entries; $A + B = B + A$ .
$kA$ : multiply every entry by $k$ .
$AB$ is defined 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 $m \times p$ and $(AB)_{ij}$ = row $i$ of $A$ times column $j$ of $B$ .
Multiplication is associative and distributes over addition; $AI = IA = A$ .
In a matrix polynomial, a constant $c$ becomes $cI$ .

$AB = \begin{bmatrix} 2 & 10 \\ -4 & 9 \end{bmatrix}$, a $2 \times 2$ matrix.
Entry of $AB$	Row of $A$ times column of $B$	Value
$(1, 1)$	$2(1) + 1(0) + 0(5)$	$2$
$(1, 2)$	$2(4) + 1(2) + 0(1)$	$10$
$(2, 1)$	$1(1) + 3(0) + (-1)(5)$	$-4$
$(2, 2)$	$1(4) + 3(2) + (-1)(1)$	$9$

Inner numbers must match; outer numbers give the order.
$A$	$B$	$AB$	$BA$
$2 \times 3$	$3 \times 2$	$2 \times 2$	$3 \times 3$
$2 \times 3$	$3 \times 4$	$2 \times 4$	not defined
$1 \times 3$	$3 \times 1$	$1 \times 1$	$3 \times 3$

Formulas

Row times column
$(AB)_{ij} = \sum_k a_{ik} b_{kj}$

Key terms

Compatible for multiplication: Columns of the first = rows of the second.

5. Non-commutativity and zero products

In general $AB \ne BA$ (one may not even be defined, or they may have different orders).
So $(A + B)^2 = A^2 + AB + BA + B^2$ and $(A + B)(A - B) = A^2 - AB + BA - B^2$ .
$AB = O$ is possible with $A \ne O$ and $B \ne O$ ; so $AB = AC$ does not give $B = C$ unless $A$ is invertible.

$A = \begin{bmatrix} 1 & 3 \\ 0 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix}$: $AB \ne BA$.
	Product
$AB$	$\begin{bmatrix} 5 & 3 \\ 2 & 2 \end{bmatrix}$
$BA$	$\begin{bmatrix} 2 & 6 \\ 1 & 5 \end{bmatrix}$

Unlike numbers, $AB = O$ does not force $A = O$ or $B = O$.
$A$	$B$	$AB$
$\begin{bmatrix} 3 & -6 \\ -1 & 2 \end{bmatrix}$	$\begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}$	$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Formulas

Expanding with care
$(A + B)^2 = A^2 + AB + BA + B^2$

Key terms

Non-commutative: $AB$ and $BA$ need not be equal.

6. Singular, non-singular and invertible matrices

A $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is singular if $ad - bc = 0$ (one row a multiple of the other) and non-singular otherwise. (The general determinant comes in the next chapter.)
A square matrix $A$ is invertible if $AB = BA = I$ for some $B$ ; then $B = A^{-1}$ . Only square matrices can be invertible; singular matrices are not.
Uniqueness: if $B$ and $C$ are inverses, $B = BI = B(AC) = (BA)C = IC = C$ .
To verify an inverse, check both $AB = I$ and $BA = I$ . To find one: solve $AB = I$ entry by entry, or use a matrix equation such as $A^2 - 3A + 2I = O \Rightarrow A^{-1} = \tfrac12 (3I - A)$ .

A zero value means singular: one row is a multiple of the other.
Matrix	$ad - bc$	Type
$\begin{bmatrix} 4 & 2 \\ 6 & 3 \end{bmatrix}$	$0$	singular
$\begin{bmatrix} 4 & 2 \\ 3 & 6 \end{bmatrix}$	$18$	non-singular
$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$	$-1$	non-singular

$A^{-1} = \begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}$; check $BA = I$ too.
Column of $B$	Equations from $AB = I$	Solution
first: $p, r$	$3p + r = 1,\; 5p + 2r = 0$	$p = 2,\; r = -5$
second: $q, s$	$3q + s = 0,\; 5q + 2s = 1$	$q = -1,\; s = 3$

A matrix equation can hand you the inverse.
Step	Result
$A^2$	$\begin{bmatrix} 7 & 3 \\ -6 & -2 \end{bmatrix}$
$A^2 - 3A + 2I$	$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
$A(3I - A)$	$\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$
$A^{-1} = \tfrac12 (3I - A)$	$\frac{1}{2}\begin{bmatrix} 0 & -1 \\ 2 & 3 \end{bmatrix}$

Formulas

Inverse
$AA^{-1} = A^{-1}A = I$

Key terms

Invertible matrix: A square matrix with an inverse.
Singular (2 × 2): $ad - bc = 0$ .

Exam tips

How ISC marks are awarded

Construct or equate (2 marks): the entries or equations, then the result.
Products (2–3 marks): each entry carries method; state the order first.
Symmetric + skew split (3–5 marks): $A'$ , $P$ , $Q$ , and the check $P + Q = A$ .
Inverse (3–5 marks): the method (equations from $AB = I$ or a matrix equation), $A^{-1}$ , and a check by multiplying back.
Section D (5 marks): set in parts such as 2 + 3 or 3 + 2; label each part.

Common mistakes

Writing the order as columns $\times$ rows.
Multiplying corresponding entries instead of rows by columns.
Assuming $AB = BA$ , or expanding $(A + B)^2$ as $A^2 + 2AB + B^2$ .
Writing a constant $c$ instead of $cI$ in a matrix equation.
Cancelling $A$ from $AB = AC$ when $A$ is not invertible.
Checking only $AB = I$ when asked to verify an inverse.

1. Order, entries and types

Formulas

Key terms

2. Equality and transpose

Formulas

Key terms

3. Symmetric and skew-symmetric matrices

Formulas

Key terms

4. Addition, scalar multiplication and multiplication

Formulas

Key terms

5. Non-commutativity and zero products

Formulas

Key terms

6. Singular, non-singular and invertible matrices

Formulas

Key terms

Exam tips

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