MathsICSE Class 10 · Matrices

Addition, Subtraction and Scalar Multiplication of Matrices (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

Matrices can be added or subtracted only if they have the same order; add or subtract the entries in matching positions. To multiply a matrix by a number k, multiply every entry by k. Build expressions like 2A − 3B + I one matrix at a time, then combine.

Add or subtract only entries in the same position; a number multiplies every entry, so 2A doubles all four.

Illustration drawn for QuestionBuddy.

Understand Addition, Subtraction and Scalar Multiplication of Matrices

These three operations are the building blocks of every longer matrix question, such as 2025 Q3(ii), which asked for $B (A + C) - 5 B$ .

Order and compatibility for addition. The order of a matrix is rows × columns. $A + B$ and $A - B$ are defined only when $A$ and $B$ have the same order. A $2 \times 2$ matrix cannot be added to a $2 \times 1$ column.

Adding and subtracting. Combine entries in the same position:

$[a c b d] \pm [p r q s] = [a \pm p c \pm r b \pm q d \pm s]$ .

Addition is commutative ( $A + B = B + A$ ) and associative ( $(A + B) + C = A + (B + C)$ ). Subtraction is not commutative: $B - A = - (A - B)$ .

Scalar multiplication. For a non-zero rational number $k$ , $k A$ multiplies every entry of $A$ by $k$ . Fractions are allowed: $\frac{1}{2} A$ halves every entry. Useful facts: $k (A + B) = k A + k B$ and $(k + m) A = k A + m A$ .

Null and identity matrices. The null matrix $O$ has every entry 0, and $A + O = A$ , $A + (- A) = O$ . The identity matrix $I = [1001]$ has 1s on the leading diagonal. So $k I = [k 0 0 k]$ , and $A + k I$ changes only the two diagonal entries of $A$ .

Mixed expressions. For something like $3 A - 2 B + I$ : write $3 A$ in full, write $2 B$ in full, then subtract and add entry by entry. The 2025 examiners reported candidates who multiplied only some entries of $B$ by 5, who found $A + B$ instead of $A + C$ , and who made slips with several operations at once. Writing each intermediate matrix separately prevents all three.

Do not confuse $2 A$ (scalar multiple) with $A^{2}$ (product $A \times A$ ).

Addition, Subtraction and Scalar Multiplication of Matrices: formulas to remember

Sum and difference: $[a c b d] \pm [p r q s] = [a \pm p c \pm r b \pm q d \pm s]$; Same order only.
Scalar multiple: $k [a c b d] = [k a k c k b k d]$; Every entry, including zeros and negatives.
Null and identity: $A + O = A, A + (- A) = O, k I = [k 0 0 k]$; $I$ changes only the diagonal when added.
Laws: $A + B = B + A, k (A + B) = k A + k B, A - B = - (B - A)$; Subtraction is not commutative.

How to solve Addition, Subtraction and Scalar Multiplication of Matrices questions

One operation per line; keep every intermediate matrix.

1Check the orders: sums and differences need the same order.
2Work out each scalar multiple ( $2 A$ , $3 B$ , $k I$ ) as a full matrix on its own line.
3If a product is involved, do brackets first, then the product, then the remaining sums.
4Combine entry by entry, keeping negative numbers in brackets.
5Read the letters again: check you used $A + C$ (not $A + B$ ) and the right multiples.

Addition, Subtraction and Scalar Multiplication of Matrices in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Sum and difference of matrices

Suppose a bookshop with two branches records fiction and non-fiction sales. Week 1: $W_{1} = [30221840]$ ; week 2: $W_{2} = [26302534]$ (rows: branches; columns: fiction, non-fiction).

Find the total sales over the two weeks and the change from week 1 to week 2.

Try it first, then show the solutionSolution

Total: $W_{1} + W_{2} = [56524374]$ .
Change: $W_{2} - W_{1} = [- 4 8 7 - 6]$ .

Answer: Totals: branch 1 sold 56 fiction and 43 non-fiction; branch 2 sold 52 and 74. Branch 1's fiction fell by 4 and its non-fiction rose by 7; branch 2's fiction rose by 8 and its non-fiction fell by 6.

Memory hook: Same shape, same labels: add or subtract box by box.

Multiplying a matrix by a rational number

Imagine a cook whose recipes for 4 people need $R = [20030010050]$ grams (rows: dish 1, dish 2; columns: rice, lentils).

Find the quantities for 6 people.

Try it first, then show the solutionSolution

6 people is $\frac{6}{4} = \frac{3}{2}$ times the recipe.
$\frac{3}{2} R = [30045015075]$ .

Answer: Dish 1: 300 g rice and 150 g lentils; dish 2: 450 g rice and 75 g lentils.

Memory hook: A scalar scales every box, never just one.

Tricks to remember Addition, Subtraction and Scalar Multiplication of Matrices

Short memory aids. Say each one aloud once, then use it on a question the same day.

Same shape to add
Lay one matrix over the other like tracing paper; if the boxes do not line up, you cannot add.
Use it when: Compatibility for addition.
The scalar visits every box
When you write $5 B$ , tick each of the four entries as you multiply it.
Use it when: Avoiding the 2025 error of multiplying only some entries.
kI is a diagonal of k
$3 I$ is 3s on the diagonal and 0s elsewhere, so $A - 3 I$ changes only the diagonal of $A$ .
Use it when: Expressions with the identity.

Addition, Subtraction and Scalar Multiplication of Matrices: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Subtracting a multiple of I · 1 mark · MCQBoard level

If $A = [20 - 1 3]$ , then $3 A - 2 I$ is:
(a) $[40 - 3 7]$
(b) $[4 - 2 - 5 7]$
(c) $[60 - 3 9]$
(d) $[4 - 2 - 3 7]$

Try it first, then show the solutionSolution

$3 A = [60 - 3 9]$ , $2 I = [2002]$ .
$3 A - 2 I = [40 - 3 7]$ . Option (b) wrongly subtracts 2 from every entry.

Answer: (a)

Example 2: Linear combinations · 3 marks · Short answerBoard level

If $A = [13 - 2 4]$ and $B = [0 - 1 52]$ , find (a) $2 A + 3 B$ (b) $A - B$ .

Try it first, then show the solutionSolution

$2 A = [26 - 4 8]$ , $3 B = [0 - 3 156]$ .
(a) $2 A + 3 B = [231114]$ .
(b) $A - B = [1 - 0 3 - (- 1) - 2 - 5 4 - 2] = [14 - 7 2]$ .

Answer: (a) $[231114]$ (b) $[14 - 7 2]$

Example 3: Fractional scalars · 3 marks · Short answerHarder board level

If $A = [46 - 2 8]$ and $B = [3 - 6 90]$ , find $\frac{1}{2} A - \frac{1}{3} B$ .

Try it first, then show the solutionSolution

$\frac{1}{2} A = [23 - 1 4]$ , $\frac{1}{3} B = [1 - 2 30]$ .
$\frac{1}{2} A - \frac{1}{3} B = [2 - 1 3 - (- 2) - 1 - 3 4 - 0] = [15 - 4 4]$ .

Answer: $[15 - 4 4]$

Example 4: Which sums are defined? · 4 marks · Short answerHarder board level

$A$ is a $2 \times 2$ matrix, $C$ is $1 \times 2$ , and $B = [3 - 1]$ , $D = [- 2 5]$ .
(a) Which of $A + B$ , $B + C$ , $B + D$ are defined? Give reasons.
(b) Find $B + D$ and $2 B - D$ .

Try it first, then show the solutionSolution

(a) $A + B$ : orders $2 \times 2$ and $2 \times 1$ differ, not defined. $B + C$ : $2 \times 1$ and $1 \times 2$ differ, not defined. $B + D$ : both $2 \times 1$ , defined.
(b) $B + D = [14]$ .
$2 B = [6 - 2]$ , so $2 B - D = [6 + 2 - 2 - 5] = [8 - 7]$ .

Answer: (a) Only $B + D$ (same order). (b) $B + D = [14]$ , $2 B - D = [8 - 7]$

Example 5: Several operations in one expression · 4 marks · Short answerHOTS / competency

$A = [2103]$ , $B = [1021]$ and $C = [- 1 2 1 - 2]$ . Find $(A + C) B - 2 B$ .

Try it first, then show the solutionSolution

$A + C = [1311]$ .
$(A + C) B = [1 + 0 3 + 0 2 + 1 6 + 1] = [1337]$ .
$2 B = [2042]$ , so $(A + C) B - 2 B = [- 1 3 - 1 5]$ .
Check another way: $(A + C - 2 I) B = [- 1 3 1 - 1] [1021] = [- 1 3 - 1 5]$ ✓.

Answer: $[- 1 3 - 1 5]$

Example 6: Case study: weighted final marks · 4 marks · Case-basedHOTS / competency

Two students' marks in English and Maths are $T_{1} = [60807550]$ in term 1 and $T_{2} = [70908565]$ in term 2 (rows: students; columns: subjects). The school's final mark is $F = \frac{2}{5} T_{1} + \frac{3}{5} T_{2}$ .
(a) Find $F$ .
(b) Find $T_{2} - T_{1}$ and say who improved most in which subject.

Try it first, then show the solutionSolution

(a) $\frac{2}{5} T_{1} = [24323020]$ , $\frac{3}{5} T_{2} = [42545139]$ .
$F = [66868159]$ .
(b) $T_{2} - T_{1} = [10101015]$ : the largest gain is 15, by student 2 in Maths.

Answer: (a) $F = [66868159]$ (b) $[10101015]$ ; student 2 in Maths.

Where students lose marks in Addition, Subtraction and Scalar Multiplication of Matrices

The slips examiners see most often, and what to write instead.

Mistake: Multiplying only some entries by the scalar
$k A$ multiplies all four entries.
Wrong working
$5 [2412] = [102012]$
Right working
$[1020510]$
Do this: Write $5 B$ as a full matrix on its own line.
Usually costs: Reported for 2025 Q3(ii).
Mistake: Using the wrong matrices
Long expressions make it easy to read $A + B$ for $A + C$ .
Wrong working
Question: $B (A + C) - 5 B$ . Student finds $A + B$ .
Right working
$A + C = [- 4 10 30]$ (for the 2025 matrices).
Do this: Copy the expression at the top of your answer and tick each letter as you use it.
Usually costs: Reported for 2025 Q3(ii); the rest of the working earns little.
Mistake: Adding kI to every entry
$I$ has 0s off the diagonal.
Wrong working
$A - 2 I$ subtracts 2 from all four entries.
Right working
$A - 2 I$ subtracts 2 from the top-left and bottom-right entries only.
Do this: Write $2 I = [2002]$ first.
Usually costs: Accuracy marks.

Exam tips

2025 Q3(ii) (4 marks) combined a sum, a product and a scalar multiple: $B (A + C) - 5 B$ . The expected steps were $A + C$ , then $B (A + C)$ , then $5 B$ , then the subtraction, each shown separately.
Keep negative entries in brackets when subtracting: $3 - (- 1) = 4$ .
State 'not defined' with the reason (orders differ) when a sum is impossible.
Check a long answer by a second route when time allows, for example $A B - 2 B = (A - 2 I) B$ .

Practice questions on Addition, Subtraction and Scalar Multiplication of Matrices

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If $A = [32 - 1 0]$ , then $A + (- A)$ is:
(a) $2 A$
(b) $O$
(c) $I$
(d) $A$
Hint
Each entry cancels.
Answer
(b) $O$ , the null matrix
Q2 · 2 marks · Short answerBoard level
If $A = [5 - 1 23]$ and $B = [23 - 4 1]$ , find $A - 2 B$ .
Hint
$2 B = [46 - 8 2]$ .
Answer
$[1 - 7 101]$
Q3 · 3 marks · Short answerHarder board level
Find $x$ and $y$ if $2 [x 0 3 y] + [14 - 2 3] = [74 4 - 1]$ .
Hint
Equate the top-left and bottom-right entries.
Answer
$2 x + 1 = 7 \Rightarrow x = 3$ ; $2 y + 3 = - 1 \Rightarrow y = - 2$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): $A - B = B - A$ for all $2 \times 2$ matrices $A$ and $B$ .
Reason (R): Addition of matrices is commutative.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
$B - A = - (A - B)$ .
Answer
(b)
Q5 · 3 marks · Short answerHarder board level
If $A = [2 - 4 68]$ , find $\frac{3}{2} A - I$ .
Hint
$\frac{3}{2} A = [3 - 6 912]$ ; subtract 1 on the diagonal only.
Answer
$[2 - 6 911]$
Q6 · 4 marks · Short answerHarder board level
Find the matrices $X$ and $Y$ if $X + Y = [5029]$ and $X - Y = [30 6 - 1]$ .
Hint
Add the equations: $2 X = \dots$
Answer
$X = [4044]$ , $Y = [10 - 2 5]$
Q7 · 4 marks · Short answerHOTS / competency
If $A = [12 - 1 0]$ and $B = [3 - 2 14]$ , find $A (B - 2 I)$ and $A B - 2 A$ . What do you notice?
Hint
$B - 2 I = [1 - 2 12]$ .
Answer
Both equal $[32 - 1 2]$ , because $A (B - 2 I) = A B - 2 A I = A B - 2 A$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.When can two matrices be added?Show answer
When they have the same order.
2.Find $[1324] + [01 - 2 1]$ .Show answer
$[1405]$
3.Find $\frac{1}{2} [4026]$ .Show answer
$[2013]$
4.Write the identity matrix of order 2.Show answer
$[1001]$
5.Is $A - B = B - A$ in general?Show answer
No; $B - A = - (A - B)$ .

Addition, Subtraction and Scalar Multiplication of Matrices: frequently asked questions

How do you add two matrices?

Check that they have the same order, then add the entries in matching positions. A 2 × 2 matrix plus a 2 × 2 matrix gives a 2 × 2 matrix.

What is scalar multiplication of a matrix?

Multiplying a matrix by a number k: every entry is multiplied by k. The ICSE syllabus includes multiplication of a 2 × 2 matrix by a non-zero rational number, so fractions such as 1/2 are allowed.

What is the difference between 2A and A²?

2A doubles every entry of A. A² is the matrix product A × A, worked out row by column. They are different in general.