Matrices · Study notes

MATTypes & Equality

Types & equality of matrices; finding unknowns

Here you classify matrices by shape and special structure (row, column, square, diagonal, scalar, identity, zero, etc.) and use the definition of equality to solve for unknown entries. Two matrices are equal only when they have the same order AND every corresponding entry matches, which converts a matrix equation into a system of ordinary equations.

Boards examine this to test precise notation and the link between matrix equality and simultaneous equations.

Equality of matrices

A = B \iff a_{ij} = b_{ij} \ \text{for all } i,j \ \text{and same order}

A=[a_{ij}]

B=[b_{ij}]

; equality requires identical order plus entrywise equality, giving one equation per position.

Order and element rule

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

m

= number of rows,

n

= number of columns; total

mn

elements.

a_{ij}

is the entry in row

i

, column

j

Diagonal / scalar / identity matrix

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \text{scalar } kI = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}

A diagonal matrix has

a_{ij}=0

for

i\ne j

; a scalar matrix is diagonal with equal diagonal entries; identity

I

is a scalar matrix with that constant

=1

Construct matrix from a rule

a_{ij} = 2i + j \Rightarrow A = \begin{pmatrix} 3 & 4 & 5 \\ 5 & 6 & 7 \end{pmatrix}

For order

2\times 3

, substitute

i=1,2

and

j=1,2,3

into the given formula to fill each position.

Equality demands SAME order first; matrices of different orders can never be equal even if entries look similar.
Equate corresponding entries to form a system of equations, then solve simultaneously for the unknowns.
Know the named types: row $(1\times n)$ , column $(m\times 1)$ , square $(m=n)$ , diagonal, scalar, identity, zero, upper/lower triangular.
A square matrix of order $n$ has $n$ diagonal entries $a_{ii}$ ; the trace is their sum.
To build a matrix from $a_{ij}=f(i,j)$ , fix the order, then run $i$ over rows and $j$ over columns.
When two unknowns appear in linked entries (e.g. $x+y$ and $x-y$ ), add/subtract the resulting equations to isolate each.
Always verify your solved values by substituting back into the original matrix entries.

Where the marks go

Comparing entries of matrices that do not even have the same order.
Confusing the index convention: $a_{ij}$ is row $i$ , column $j$ — students often swap row and column.
Solving only one of the simultaneous equations and stopping, instead of finding all unknowns.
Mislabelling a scalar matrix as identity (identity needs the constant to be exactly 1).

How the board asks it

Numericalequating corresponding entries to form simultaneous equations
If $\begin{bmatrix} x+3 & 2y+x \\ z-1 & 4w-6 \end{bmatrix} = \begin{bmatrix} 0 & -7 \\ 3 & 2w \end{bmatrix}$ , find the values of $x$ , $y$ , $z$ and $w$ .
Numericaladd or subtract linked entries such as $x+y$ and $x-y$
Given that $\begin{bmatrix} x+y & 2 \\ 5 & x-y \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 2 \end{bmatrix}$ , find the values of $x$ and $y$ .
Numericalbuild a matrix from $a_{ij}=f(i,j)$
Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by $a_{ij} = \dfrac{(i+2j)^2}{2}$ .
Identify / classifynamed types: row, column, diagonal, scalar, identity
Define a scalar matrix and explain, with reasons, why every identity matrix is a scalar matrix but not every scalar matrix is an identity matrix.
Distinguishequality demands the same order first
A matrix $A$ has $12$ elements. Write all possible orders it can have. If instead $A$ and $B$ have orders $2 \times 3$ and $3 \times 2$ but contain the same six numbers, state with a reason whether $A = B$ is possible.

Practise this topic

NextMatrix operations (addition, scalar multiplication, multiplication)

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.