Matrices and Determinants of Order 2

Lessons

Senior 4 · Mathematics

Matrices and Determinants of Order 2

5 lessons

Lesson 2 of 5

Equality, addition and scalar multiplication of matrices

25 min

Tables that can be added

A small coffee roaster in Kigali sells roasted beans in $250\,\mathrm{g}$ and $500\,\mathrm{g}$ bags through two shops. The numbers of bags sold in two weeks were (illustrative figures):

Week 1	250 g bags	500 g bags
Shop 1	40	25
Shop 2	32	18

Week 2	250 g bags	500 g bags
Shop 1	36	30
Shop 2	35	20

Stripped of their labels, the tables are the matrices

P = \begin{pmatrix} 40 & 25 \\ 32 & 18 \end{pmatrix}, \qquad Q = \begin{pmatrix} 36 & 30 \\ 35 & 20 \end{pmatrix}.

The total for the fortnight is found by adding entries in matching places, and the change from week 1 to week 2 by subtracting them. If every week of the next month repeats week 2, each entry is multiplied by 4. These are the three operations of this lesson.

Definition

Matrix of order $m \times n$ : a rectangular array of real numbers with $m$ rows and $n$ columns. The entry in row $i$ , column $j$ of $A$ is $a_{ij}$ . A $2 \times 2$ matrix is square of order 2; a $2 \times 1$ matrix is a column vector.

Equal matrices: $A = B$ when $A$ and $B$ have the same order and $a_{ij} = b_{ij}$ for every $i$ and $j$ .

Example: $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \ne \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$ : the orders $2 \times 2$ and $2 \times 3$ differ.

Definition

Sum, scalar multiple and difference: for matrices $A = (a_{ij})$ and $B = (b_{ij})$ of the same order and a real number $k$ ,

A + B = (a_{ij} + b_{ij}), \qquad kA = (k\,a_{ij}), \qquad A - B = A + (-1)B = (a_{ij} - b_{ij}).

The zero matrix $O$ has every entry 0, and the opposite of $A$ is $-A = (-a_{ij})$ .

Example: $P + Q = \begin{pmatrix} 76 & 55 \\ 67 & 38 \end{pmatrix}$ , $Q - P = \begin{pmatrix} -4 & 5 \\ 3 & 2 \end{pmatrix}$ and $4Q = \begin{pmatrix} 144 & 120 \\ 140 & 80 \end{pmatrix}$ . The entry $- 4$ says Shop 1 sold 4 fewer small bags in week 2.

Worked example 1: using equality

Find $x$ , $y$ and $z$ if $\begin{pmatrix} x + y & 3 \\ 2z & x - y \end{pmatrix} = \begin{pmatrix} 7 & 3 \\ 10 & 1 \end{pmatrix}$ .

Solution. Equal matrices have equal entries in each place: $x + y = 7$ , $x - y = 1$ and $2 z = 10$ . Adding the first two, $2 x = 8$ , so $x = 4$ and $y = 3$ ; and $z = 5$ .

Worked example 2: combining operations

Let $A = \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} -4 & 5 \\ 1 & 2 \end{pmatrix}$ . Find $2 A - 3 B$ , and solve $2 X + A = B$ for the matrix $X$ .

Solution.

2A - 3B = \begin{pmatrix} 4 & -2 \\ 0 & 6 \end{pmatrix} - \begin{pmatrix} -12 & 15 \\ 3 & 6 \end{pmatrix} = \begin{pmatrix} 16 & -17 \\ -3 & 0 \end{pmatrix}.

From $2 X + A = B$ : $2 X = B - A$ , so

X = \tfrac{1}{2}(B - A) = \tfrac{1}{2}\begin{pmatrix} -6 & 6 \\ 1 & -1 \end{pmatrix} = \begin{pmatrix} -3 & 3 \\ 0.5 & -0.5 \end{pmatrix}.

Check: $2X + A = \begin{pmatrix} -6 & 6 \\ 1 & -1 \end{pmatrix} + \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} -4 & 5 \\ 1 & 2 \end{pmatrix} = B$ . ✓

The $2 \times 2$ matrices form a vector space

Matrix addition and scalar multiplication work entry by entry, exactly like the componentwise operations on $\mathbb{R}^2$ , so they obey the same ten rules. Writing $M_2(\mathbb{R})$ for the set of all $2 \times 2$ real matrices, for all $A, B, C \in M_2(\mathbb{R})$ and all real $k, m$ :

Rule	Statement
commutative, associative	$A + B = B + A$ , $(A + B) + C = A + (B + C)$
zero and opposite	$A + O = A$ , $A + (- A) = O$
distributive	$k (A + B) = k A + k B$ , $(k + m) A = k A + m A$
scalars	$k (m A) = (k m) A$ , $1 A = A$

So $M_2(\mathbb{R})$ is a vector space. Its four "standard" matrices

E_{11} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\ E_{12} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\ E_{21} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},\ E_{22} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}

form a basis, because $\begin{pmatrix} a & b \\ c & d \end{pmatrix} = aE_{11} + bE_{12} + cE_{21} + dE_{22}$ in exactly one way. Hence $\dim M_2(\mathbb{R}) = 4$ .

What the sum means for transformations

If $f$ and $g$ are linear with matrices $A$ and $B$ , define $(f + g)(\vec{v}) = f(\vec{v}) + g(\vec{v})$ and $(kf)(\vec{v}) = k f(\vec{v})$ . Then $(f + g)(\vec{v}) = A\vec{v} + B\vec{v} = (A + B)\vec{v}$ , so the matrix of $f + g$ is $A + B$ , and the matrix of $k f$ is $k A$ . (The middle step, $A\vec{v} + B\vec{v} = (A + B)\vec{v}$ , is one line of algebra: compare the entries.)

Worked example 3: the sum of two transformations

Let $f$ be the reflection in the $x$ -axis and $g$ the reflection in the line $y = x$ . Find the matrix of $f + g$ and the image of $(2, 1)$ under it.

Solution.

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \qquad \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}.

Check directly: $f (2, 1) = (2, - 1)$ and $g (2, 1) = (1, 2)$ ; their sum is $(3, 1)$ . ✓ Note that $f + g$ is not itself a reflection: adding two transformations adds their output vectors; it does not do one after the other.

James Joseph Sylvester (1814–1897), painted by Alfred Edward Emslie in 1889. Sylvester was the first to use the word "matrix" for a rectangular array of numbers, in 1850. Painting: Alfred Edward Emslie, public domain, via Wikimedia Commons.

Warning

Same order, or no sum at all. $\begin{pmatrix} 1 & 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ is undefined: a $1 \times 2$ and a $2 \times 1$ matrix cannot be added, even though both hold the numbers 1 and 2. And a scalar multiplies every entry: $3\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix}$ , not just the first row.

Tip

Solve a matrix equation such as $3 X - A = B$ exactly like the number equation $3 x - a = b$ : move $A$ across, then multiply by $\frac{1}{3}$ . Do every step with whole matrices before you calculate any entry.

Mathispower4u: Matrix addition, subtraction and scalar multiplication (about 10 min). Watch for the check on the orders before any matrices are added.

✦

Practice

Work each one, then open the answers.

With $A = \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} -4 & 5 \\ 1 & 2 \end{pmatrix}$ , find (a) $A + B$ (b) $A - B$ (c) $3 A$ (d) $2 A - B$ .
Find $x$ and $y$ if $\begin{pmatrix} 2x & 5 \\ y + 1 & 4 \end{pmatrix} = \begin{pmatrix} 6 & 5 \\ -3 & 4 \end{pmatrix}$ .
Find $a, b, c, d$ if $\begin{pmatrix} a + b & 2 \\ c & a - b \end{pmatrix} = \begin{pmatrix} 9 & 2c - 4 \\ d & 1 \end{pmatrix}$ .
Solve $3X - \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 5 & -2 \\ 0 & 2 \end{pmatrix}$ .
Prove that $k (A + B) = k A + k B$ for all $2 \times 2$ matrices $A, B$ and every real $k$ .
Write $M = \begin{pmatrix} 3 & -1 \\ 4 & 0 \end{pmatrix}$ as a linear combination of $E_{11}, E_{12}, E_{21}, E_{22}$ .
$f$ is the reflection in the $x$ -axis and $g$ the reflection in the $y$ -axis. Find the matrix of $f + g$ and describe $f + g$ .
The matrix $C = \begin{pmatrix} 600 & 900 \\ 650 & 850 \end{pmatrix}$ gives the prices in Frw of 1 kg of beans (column 1) and 1 kg of rice (column 2) at two markets (illustrative). Every price rises by 10%. Find the new price matrix.

Show answersHide answers

(a) $\begin{pmatrix} -2 & 4 \\ 1 & 5 \end{pmatrix}$ (b) $\begin{pmatrix} 6 & -6 \\ -1 & 1 \end{pmatrix}$ (c) $\begin{pmatrix} 6 & -3 \\ 0 & 9 \end{pmatrix}$ (d) $\begin{pmatrix} 8 & -7 \\ -1 & 4 \end{pmatrix}$ .
$2 x = 6$ , so $x = 3$ ; $y + 1 = - 3$ , so $y = - 4$ .
$a + b = 9$ and $a - b = 1$ give $a = 5$ , $b = 4$ ; $2 = 2 c - 4$ gives $c = 3$ ; $d = c = 3$ .
$3X = \begin{pmatrix} 6 & 0 \\ 3 & 6 \end{pmatrix}$ , so $X = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}$ .
The entry in place $(i, j)$ of $k (A + B)$ is $k(a_{ij} + b_{ij}) = ka_{ij} + kb_{ij}$ , which is the entry of $k A + k B$ in the same place.
$M = 3E_{11} - E_{12} + 4E_{21} + 0E_{22}$ .
$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} + \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = O$ : $f + g$ sends every vector to $\vec{0}$ (the zero transformation).
$1.1C = \begin{pmatrix} 660 & 990 \\ 715 & 935 \end{pmatrix}$ .

Quick check

Key points

Equal matrices have the same order and equal entries in every place.
Add, subtract and multiply by a scalar entry by entry; only matrices of the same order can be added.
$M_2(\mathbb{R})$ is a vector space of dimension 4, with basis $E_{11}, E_{12}, E_{21}, E_{22}$ .
The matrix of $f + g$ is $A + B$ , and the matrix of $k f$ is $k A$ .

Sources and credits

Written by Ganzaa: the explanations, the worked examples (the sales and price figures are illustrative), the practice items and the questions. The unit follows the topic list of REB's S4 Core Mathematics (2026–2027); no REB text is reproduced.
History: Matrices and determinants, J. J. O'Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews.
Photo: Rwandan coffee beans, Evan-Amos, Wikimedia Commons, CC0 1.0.
Painting: James Joseph Sylvester, Alfred Edward Emslie, Wikimedia Commons, public domain.
Video: Matrix Addition, Subtraction, and Scalar Multiplication, Mathispower4u, embedded from YouTube.

1Tables that can be added

2The 2×2 2 \times 2 2×2 matrices form a vector space

3What the sum means for transformations

Tables that can be added

The $2 \times 2$ matrices form a vector space

What the sum means for transformations