Matrices: Types, Multiplication and Transpose | NIMCET

Quick answer: A matrix of order $m \times n$ is a rectangular array with m rows and n columns. Matrices of the same order are added entry by entry. The product AB exists only when the number of columns of A equals the number of rows of B, and in general $AB \ne BA$ . The transpose $A^T$ swaps rows and columns, with $(AB)^T = B^TA^T$ . A is symmetric if $A^T = A$ and skew-symmetric if $A^T = -A$ .

Types of matrices

Type	Definition
Row / column	One row / one column
Square	$m = n$
Zero (null)	All entries 0
Diagonal	Square, all off-diagonal entries 0
Scalar	Diagonal with equal diagonal entries
Identity $I$	Diagonal with all diagonal entries 1
Upper triangular	All entries below the diagonal are 0
Lower triangular	All entries above the diagonal are 0

Trace: the sum of the diagonal entries of a square matrix. $\text{tr}(A + B) = \text{tr}(A) + \text{tr}(B)$ and $\text{tr}(AB) = \text{tr}(BA)$ .

Matrix operations

Operation	Rule
Addition	Same order; add corresponding entries
Scalar multiple	Multiply every entry by k
Product AB	$(AB)_{ij} = \sum_k a_{ik}b_{kj}$ ; needs columns of A = rows of B
Order of product	$(m \times n)(n \times p) = m \times p$

Properties of multiplication

Property	Holds?
Associative: $(AB)C = A(BC)$	Yes
Distributive: $A(B + C) = AB + AC$	Yes
Commutative: $AB = BA$	Not in general
$AB = O \Rightarrow A = O$ or $B = O$	No
$AI = IA = A$	Yes

Because $AB \ne BA$ in general, $(A + B)^2 = A^2 + AB + BA + B^2$ , not $A^2 + 2AB + B^2$ .

Transpose

Property	Result
$(A^T)^T$	$A$
$(A + B)^T$	$A^T + B^T$
$(kA)^T$	$kA^T$
$(AB)^T$	$B^TA^T$

Symmetric and skew-symmetric matrices

Symmetric: $A^T = A$ , so $a_{ij} = a_{ji}$ .
Skew-symmetric: $A^T = -A$ , so $a_{ij} = -a_{ji}$ and all diagonal entries are 0.

Every square matrix splits uniquely as

$A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$

where the first part is symmetric and the second skew-symmetric.

Special square matrices

Name	Condition
Idempotent	$A^2 = A$
Involutory	$A^2 = I$
Nilpotent of index k	$A^k = O$ , $A^{k - 1} \ne O$
Orthogonal	$AA^T = A^TA = I$

Worked examples

Example 1: multiplication

Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ .

$AB = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}, \quad BA = \begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix}$

So $AB \ne BA$ .

Example 2: order of a product

If A is $2 \times 3$ and B is $3 \times 4$ , then AB is $2 \times 4$ , and BA is not defined.

Example 3: symmetric part

Write $A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}$ as symmetric plus skew-symmetric.

$\frac{1}{2}(A + A^T) = \begin{bmatrix} 1 & 3 \\ 3 & 3 \end{bmatrix}, \quad \frac{1}{2}(A - A^T) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

Example 4: powers of a matrix

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ , then $A^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$ . Check $n = 2$ : $A^2 = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ ✓.

Summary table

Idea	Key result
Product defined	Columns of A = rows of B
Commutativity	$AB \ne BA$ in general
Transpose of product	$B^TA^T$
Skew-symmetric diagonal	All zeros
Any square matrix	Symmetric + skew-symmetric

Exam tips

Check orders first; many questions are answered by "not defined".
Do not expand $(A + B)^2$ as if matrices commute.
For $A^n$ , compute $A^2$ and $A^3$ and look for a pattern.

Next, study determinants, cofactors and the inverse, or see the NIMCET Algebra chapter.

Common mistakes

Multiplying matrices entry by entry.
Writing $(AB)^T = A^TB^T$ .
Assuming $AB = O$ forces A or B to be zero.
Forgetting that a skew-symmetric matrix has zero diagonal.

Frequently Asked Questions

When can two matrices be multiplied?

AB is defined when the number of columns of A equals the number of rows of B.

Is matrix multiplication commutative?

No. In general AB ≠ BA.

What is the transpose of a product of matrices?

(AB)ᵀ = BᵀAᵀ.

What is a skew-symmetric matrix?

A square matrix with Aᵀ = −A. All its diagonal entries are zero.

What is an idempotent matrix?

A square matrix with A² = A.