Matrix operations (addition, scalar multiplication, multiplication)

MATMatrix Operations

This subtopic covers the algebra of matrices: adding/subtracting matrices of the same order, multiplying every entry by a scalar, and the row-by-column product of two conformable matrices. Addition and scalar multiplication act entrywise, while multiplication uses dot products of rows with columns and is generally non-commutative.

These operations are the foundation for every later topic (transpose laws, inverses, matrix equations), so accuracy with order and dimensions is essential.

Matrix addition (same order)

(A + B)_{ij} = a_{ij} + b_{ij}

A=[a_{ij}]

and

B=[b_{ij}]

must have the SAME order

m\times n

; the sum has order

m\times n

Scalar multiplication

(kA)_{ij} = k\,a_{ij}

k

is a scalar; every entry is multiplied by

k

. Combine with addition for expressions like

2A-3B

Matrix multiplication (row by column)

(AB)_{ij} = \sum_{k=1}^{n} a_{ik}\,b_{kj}

Defined only if columns of

A

(

=n

) equal rows of

B

; entry

(i,j)

is the dot product of row

i

A

with column

j

B

Order of a product

A_{m \times n}\,B_{n \times p} = C_{m \times p}

Inner dimensions must match (

n=n

); the product inherits outer dimensions

m\times p

Reversal law for transpose of a product

(AB)' = B'A'

A'

(or

A^{T}

) is the transpose; note the ORDER reverses. Used to verify computed products.

Add or subtract matrices only when their orders are identical; otherwise the operation is undefined.
For $kA$ , multiply EVERY entry by $k$ , including zeros; for $2A-3B$ scale each first, then subtract entrywise.
For $AB$ , check the inner dimensions match before computing; the result order is rows-of- $A$ by columns-of- $B$ .
Each product entry is one dot product: $(i,j)$ entry = row $i$ of $A$ dotted with column $j$ of $B$ .
Matrix multiplication is associative $(AB)C=A(BC)$ and distributive over addition, but NOT commutative in general.
$AB$ may exist while $BA$ does not (or has a different order); always state the order of your answer.
$AB=O$ does not imply $A=O$ or $B=O$ — zero divisors exist for matrices.

Where the marks go

Trying to add matrices of different orders, or assuming $AB=BA$ .
Multiplying only the diagonal or doing entrywise multiplication instead of the row-by-column rule.
Forgetting that $(AB)'=B'A'$ reverses the order — writing $A'B'$ is wrong.
Mis-stating the order of the product (e.g. giving the order of $A$ instead of rows-of- $A$ by columns-of- $B$ ).

How the board asks it

Numericalmatrix multiplication (row by column) and order of a product
If $A=\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ and $B=\begin{bmatrix}2 & 0\\1 & 3\end{bmatrix}$ , find the product $AB$ and state its order.
Numericalscalar multiplication and entrywise subtraction
Given $A=\begin{bmatrix}2 & -1\\0 & 3\end{bmatrix}$ and $B=\begin{bmatrix}1 & 4\\-2 & 5\end{bmatrix}$ , evaluate $2A-3B$ .
Conversionequality of matrices of the same order
Find the values of $x$ and $y$ such that $\begin{bmatrix}x+y & 2\\5 & x-y\end{bmatrix}=\begin{bmatrix}6 & 2\\5 & 2\end{bmatrix}$ .
Derive / provenon-commutativity of matrix multiplication
For $A=\begin{bmatrix}1 & 2\\3 & 1\end{bmatrix}$ and $B=\begin{bmatrix}0 & 1\\1 & 2\end{bmatrix}$ , compute $AB$ and $BA$ and hence show that $AB\neq BA$ .
Give reasonszero divisors: $AB=O$ does not imply $A=O$ or $B=O$
Give an example of two non-zero $2\times 2$ matrices $A$ and $B$ such that $AB=O$ , and state what this illustrates about matrix multiplication.
Applicationconformability and revenue as a row-by-column product
A shop sells pens and notebooks. The quantities sold are $Q=\begin{bmatrix}40 & 25\end{bmatrix}$ and the prices in rupees are $P=\begin{bmatrix}5\\30\end{bmatrix}$ ; find the total revenue $QP$ and state the order of each matrix.

Practise this topic

PreviousTypes & equality of matrices; finding unknowns NextMatrix equations / solving for X

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.