Operations on Matrices

Addition and scalar multiplication

Two matrices can be added only if they have the same order; add entry by entry. Multiplying by a scalar multiplies every entry:

\begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} a+e & b+f \\ c+g & d+h \end{bmatrix}, \qquad k\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} ka & kb \\ kc & kd \end{bmatrix}

Multiplication

$AB$ is defined when the number of columns of $A$ equals the number of rows of $B$ . An $m \times n$ matrix times an $n \times p$ matrix gives an $m \times p$ matrix, and each entry is the sum of products of a row of $A$ with a column of $B$ .

Worked example

Given:

A = \begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & -1 \\ 2 & 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 2 \\ 1 & 1 & -2 \end{bmatrix}, \quad C = \begin{bmatrix} 4 \\ -1 \\ 2 \end{bmatrix}

Calculate $AB$ and $AC$ . ( $B$ is 3×3 and $C$ is 3×1, so $B + C$ isn't defined.)

AB = \begin{bmatrix} 7 & -3 & -9 \\ -5 & 11 & 10 \\ 3 & 3 & 4 \end{bmatrix}

For example, the top-left entry is row 1 of $A$ times column 1 of $B$ : $(1)(2) + (-2)(-1) + (3)(1) = 7$ .

AC = \begin{bmatrix} 1(4) + (-2)(-1) + 3(2) \\ 0(4) + 4(-1) + (-1)(2) \\ 2(4) + 1(-1) + 0(2) \end{bmatrix} = \begin{bmatrix} 12 \\ -6 \\ 7 \end{bmatrix}

Worked example: AB and BA differ

Let $P = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $Q = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$ .

PQ = \begin{bmatrix} 7 & 2 \\ 3 & 1 \end{bmatrix}, \qquad QP = \begin{bmatrix} 1 & 2 \\ 3 & 7 \end{bmatrix}

So $PQ \ne QP$ : matrix multiplication isn't commutative. Also, a product can be zero without either factor being zero: $Z = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ is not the zero matrix, but $Z^2 = O$ .

Useful properties

$A(BC) = (AB)C$ and $A(B + C) = AB + AC$ , whenever the products are defined.
$(AB)^T = B^TA^T$ , with the order reversed. For $A$ and $B$ above, this holds entry by entry.
$AI = IA = A$ for the identity matrix of matching order.

Common mistakes

Assuming $AB = BA$ . It usually doesn't, as $PQ$ and $QP$ show.
Multiplying entry by entry. That's not matrix multiplication; use rows times columns.
Cancelling a matrix. $AB = AC$ doesn't imply $B = C$ unless $A$ is invertible.
Writing $(AB)^T = A^TB^T$ . The order reverses.

Where this comes up in exams

The 2025-26 CBSE Class XII syllabus lists operations on matrices (addition, multiplication and multiplication by a scalar), the non-commutativity of multiplication and non-zero matrices whose product is the zero matrix. JEE Main 2026 lists the algebra of matrices, and JEE Advanced 2026 lists addition, multiplication by a scalar and product of matrices, and the transpose.

Checked against CBSE Mathematics curriculum, Classes XI-XII (2025-26), JEE (Main) 2026 syllabus (NTA) and JEE (Advanced) 2026 syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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