MathsICSE Class 10 · Matrices

Multiplication of Matrices: Row-by-Column Rule, A² and Solved Questions (ICSE Class 10)

Q: How do you multiply two 2 × 2 matrices?

Each entry of the product is a row of the first matrix times a column of the second: multiply the pairs and add. For example, the top-left entry is (row 1 of A) times (column 1 of B).

Q: Why is A² not found by squaring each element?

Because A² means A × A, and matrix multiplication combines rows with columns. Squaring each element gives a different matrix in almost every case; CISCE examiners reported this error in 2024 and 2025.

Q: Which matrix products does the ICSE syllabus include?

The CISCE syllabus covers compatibility for multiplication and multiplication of a 2 × 2 matrix by a matrix, so you meet 2 × 2 by 2 × 2, 2 × 2 by a column, and a row by 2 × 2.

Updated 15 October 2026 · Fact-checked

To multiply matrices, multiply each row of the first matrix by each column of the second, entry by entry, and add. AB is defined only when the columns of A equal the rows of B. A² means A × A, not squaring each entry, and in general AB ≠ BA.

Pair row 1 of A with column 1 of B: first with first, second with second, then add to get the top-left entry of AB.

Illustration drawn for QuestionBuddy.

Understand Multiplication of Matrices

Matrix multiplication is the operation ICSE tests most often in this chapter: it appeared in a 1-mark MCQ in both 2024 and 2025 and inside the 4-mark questions too.

When is AB defined? If $A$ has order $m \times n$ and $B$ has order $n \times p$ , the product $A B$ exists and has order $m \times p$ . The inner numbers must match; the outer numbers give the answer's order. In ICSE you meet three cases:

$2 \times 2$ times $2 \times 2$ gives $2 \times 2$ .
$2 \times 2$ times $2 \times 1$ (a column) gives $2 \times 1$ .
$1 \times 2$ (a row) times $2 \times 2$ gives $1 \times 2$ .

A column times a $2 \times 2$ matrix ( $2 \times 1$ by $2 \times 2$ ) is not defined.

The row-by-column rule. For $A = [a c b d]$ and $B = [p r q s]$ :

$A B = [a p + b r c p + d r a q + b s c q + d s]$ .

The entry in row $i$ , column $j$ of $A B$ comes from row $i$ of $A$ and column $j$ of $B$ : multiply first with first, second with second, then add.

Squares and powers. $A^{2} = A \times A$ and $A^{3} = A^{2} \times A$ . The examiners reported in both 2024 Q1(iv) and 2025 Q1(v) that many candidates squared each entry instead; that gives the wrong matrix except in special cases.

Order matters. In general $A B \neq = B A$ , so matrix multiplication is not commutative. Consequences: $(A + B)^{2} = A^{2} + A B + B A + B^{2}$ , not $A^{2} + 2 A B + B^{2}$ , and $(A + B) (A - B)$ need not equal $A^{2} - B^{2}$ . Multiplication is still associative, $(A B) C = A (B C)$ , and distributes over addition, $A (B + C) = A B + A C$ .

Identity and null matrices. $A I = I A = A$ and $A O = O A = O$ , just like multiplying by 1 and 0. Unlike numbers, $A B = O$ does not force $A = O$ or $B = O$ .

Multiplying by a number (scalar multiplication) is different: $k A$ multiplies every entry by $k$ . Do not confuse $2 A$ with $A^{2}$ .

Multiplication of Matrices: formulas to remember

Product of two 2 × 2 matrices: $[a c b d] [p r q s] = [a p + b r c p + d r a q + b s c q + d s]$; Row of the first times column of the second.
2 × 2 matrix times a column: $[a c b d] [x y] = [a x + b y c x + d y]$; Order 2 × 1.
Row times a 2 × 2 matrix: $[x y] [a c b d] = [a x + cy b x + d y]$; Order 1 × 2.
Order of a product: $(m \times n) (n \times p) = m \times p$; Defined only when the inner numbers match.
Powers and identity: $A^{2} = AA, A^{3} = A^{2} A, A I = I A = A$; Never square entries one by one.

How to solve Multiplication of Matrices questions

Check the orders first, then work one entry at a time and write each sum before simplifying.

1Write the orders: is the number of columns of the first equal to the number of rows of the second? If not, the product is not defined.
2Write the order of the answer (outer numbers) and draw an empty matrix of that size.
3For each entry, take the matching row of the first and column of the second; write the sum such as $2 \times 1 + 3 \times 3$ in full.
4Simplify each sum, watching signs of negative entries.
5For $A^{2}$ , write $A \times A$ and multiply; for $A^{3}$ , multiply $A^{2}$ by $A$ .
6If asked whether $A B = B A$ , find both products separately and compare entry by entry.

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.

Row-by-column multiplication as a total

Suppose two teams in a school football league have these results: Team P won 5 and drew 2; Team Q won 3 and drew 4. A win earns 3 points and a draw 1 point.

Write the results as a $2 \times 2$ matrix and the points as a column matrix, and find each team's points by multiplication.

Try it first, then show the solutionSolution

Results $R = [5324]$ (rows: teams; columns: wins, draws); points $P = [31]$ .
$R P = [5 \times 3 + 2 \times 1 3 \times 3 + 4 \times 1] = [1713]$ .

Answer: Team P has 17 points and Team Q has 13 points.

Memory hook: Row × column = 'how many' times 'how much each', added up.

Product of two 2 × 2 matrices

Imagine a baker with two recipes: recipe 1 needs 2 kg flour and 1 kg sugar per batch; recipe 2 needs 3 kg flour and 2 kg sugar. Shop X charges ₹40 per kg of flour and ₹50 per kg of sugar; shop Y charges ₹45 and ₹44.

Use a matrix product to find the cost of one batch of each recipe at each shop, and say where each recipe is cheaper.

Try it first, then show the solutionSolution

Recipes $R = [2312]$ (rows: recipes; columns: flour, sugar). Prices $C = [40504544]$ (rows: flour, sugar; columns: shops X, Y).
$R C = [80 + 50 120 + 100 90 + 44 135 + 88] = [130220134223]$ .
Compare along each row: 130 < 134 and 220 < 223.

Answer: Recipe 1 costs ₹130 at X and ₹134 at Y; recipe 2 costs ₹220 at X and ₹223 at Y. Both are cheaper at shop X.

Memory hook: The middle labels must match (flour, sugar); the outer labels (recipes, shops) name the answer.

Tricks to remember Multiplication of Matrices

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

Run across, dive down
Your left finger runs across a row of the first matrix while your right finger dives down a column of the second; multiply the pairs and add.
Use it when: Getting every entry of a product right.
Inner numbers match, outer numbers stay
$(2 \times 2) (2 \times 1)$ : the bold inner pair must be equal; the outer pair $2 \times 1$ is the answer's order.
Use it when: Compatibility and order questions.
A² is a product, not a power of entries
Say 'A squared is A times A' aloud before you start.
Use it when: Avoiding the error reported in 2024 Q1(iv) and 2025 Q1(v).

Multiplication of Matrices: solved examples

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

Example 1: Squaring a matrix · 1 mark · MCQBoard level

If $A = [3103]$ , then $A^{2}$ is:
(a) $[9109]$
(b) $[9609]$
(c) $[9309]$
(d) $[6206]$

Try it first, then show the solutionSolution

$A^{2} = [3103] [3103]$ .
Row 1: $3 \times 3 + 0 \times 1 = 9$ , $3 \times 0 + 0 \times 3 = 0$ .
Row 2: $1 \times 3 + 3 \times 1 = 6$ , $1 \times 0 + 3 \times 3 = 9$ .
Option (a) comes from squaring entries; (d) is $2 A$ .

Answer: (b) $[9609]$

Example 2: AB and BA · 3 marks · Short answerBoard level

If $A = [2 - 1 34]$ and $B = [13 - 2 0]$ , find $A B$ and $B A$ . Is $A B = B A$ ?

Try it first, then show the solutionSolution

$A B = [2 \times 1 + 3 \times 3 - 1 \times 1 + 4 \times 3 2 \times (- 2) + 3 \times 0 (- 1) (- 2) + 4 \times 0] = [1111 - 4 2]$ .
$B A = [1 \times 2 + (- 2) (- 1) 3 \times 2 + 0 \times (- 1) 1 \times 3 + (- 2) \times 4 3 \times 3 + 0 \times 4] = [46 - 5 9]$ .
The entries differ, so $A B \neq = B A$ .

Answer: $A B = [1111 - 4 2]$ , $B A = [46 - 5 9]$ ; $A B \neq = B A$ .

Example 3: Rows and columns with a 2 × 2 matrix · 4 marks · Short answerHarder board level

Let $M = [4215]$ , $P = [2 - 1]$ and $Q = [3 - 2]$ .
(a) Find $M P$ and $QM$ .
(b) Explain why $P M$ is not defined.

Try it first, then show the solutionSolution

$M$ is $2 \times 2$ and $P$ is $2 \times 1$ : inner numbers 2 and 2 match, so $M P$ is $2 \times 1$ .
$M P = [4 \times 2 + 1 \times (- 1) 2 \times 2 + 5 \times (- 1)] = [7 - 1]$ .
$Q$ is $1 \times 2$ , so $QM$ is $1 \times 2$ : $QM = [3 \times 4 + (- 2) \times 2 3 \times 1 + (- 2) \times 5] = [8 - 7]$ .
(b) $P$ is $2 \times 1$ and $M$ is $2 \times 2$ : $P$ has 1 column but $M$ has 2 rows, so $P M$ is not defined.

Answer: (a) $M P = [7 - 1]$ , $QM = [8 - 7]$ (b) columns of $P$ (1) ≠ rows of $M$ (2).

Example 4: (A + B)(A − B) is not A² − B² · 4 marks · Short answerHarder board level

If $A = [2110]$ and $B = [0111]$ , find $(A + B) (A - B)$ and $A^{2} - B^{2}$ . Explain why they differ.

Try it first, then show the solutionSolution

$A + B = [2221]$ , $A - B = [20 0 - 1]$ .
$(A + B) (A - B) = [4 + 0 4 + 0 0 - 2 0 - 1] = [44 - 2 - 1]$ .
$A^{2} = [5221]$ , $B^{2} = [1112]$ , so $A^{2} - B^{2} = [41 1 - 1]$ .
Expanding, $(A + B) (A - B) = A^{2} - A B + B A - B^{2}$ ; the middle terms cancel only if $A B = B A$ , which is not true here.

Answer: $(A + B) (A - B) = [44 - 2 - 1]$ , $A^{2} - B^{2} = [41 1 - 1]$ ; they differ because $A B \neq = B A$ .

Example 5: Spotting a pattern in powers · 4 marks · Short answerHOTS / competency

If $A = [1021]$ , find $A^{2}$ and $A^{3}$ . Hence write down $A^{10}$ .

Try it first, then show the solutionSolution

$A^{2} = [1 \times 1 + 2 \times 0 0 1 \times 2 + 2 \times 1 1] = [1041]$ .
$A^{3} = A^{2} A = [10 2 + 4 1] = [1061]$ .
Each multiplication by $A$ adds 2 to the top-right entry, so $A^{n} = [10 2 n 1]$ .

Answer: $A^{2} = [1041]$ , $A^{3} = [1061]$ , $A^{10} = [10201]$ .

Example 6: Case study: takings and profit at two stalls · 4 marks · Case-basedHOTS / competency

Two school-fair stalls sell pens and notebooks. Stall 1 sells 20 pens and 15 notebooks; stall 2 sells 10 pens and 25 notebooks. A pen sells for ₹10 (cost ₹8) and a notebook for ₹40 (cost ₹30).
(a) Write the sales matrix $S$ and a price matrix $P$ whose two columns are selling price and cost price.
(b) Find $S P$ .
(c) Find each stall's profit.

Try it first, then show the solutionSolution

(a) $S = [20101525]$ (rows: stalls; columns: pens, notebooks); $P = [1040830]$ (rows: pens, notebooks; columns: selling, cost).
(b) $S P = [200 + 600 100 + 1000 160 + 450 80 + 750] = [8001100610830]$ .
(c) Column 1 is takings, column 2 is cost: profit $= 800 - 610 = 190$ and $1100 - 830 = 270$ .

Answer: (b) $[8001100610830]$ (c) Stall 1: ₹190; stall 2: ₹270.

Where students lose marks in Multiplication of Matrices

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

Mistake: Squaring each entry to get A²
$A^{2}$ is the product $A \times A$ , found row by column.
Wrong working
$[2022]^{2} = [4044]$
Right working
$[2022] [2022] = [4084]$
Do this: Write $A^{2} = A \times A$ as your first line.
Usually costs: The whole mark in 2024 Q1(iv) and 2025 Q1(v), where the examiners reported this error.
Mistake: Multiplying entry by entry in the same position
Matrix multiplication pairs a row with a column, not matching positions.
Wrong working
$[x 1 01] [4 y 01] = [4 x y 01]$
Right working
Row 2 times column 1 is $1 \times 4 + 1 \times y = 4 + y$ .
Do this: Say 'row 2, column 1' for each entry and use the whole row and column.
Usually costs: Reported in 2024 Q2(i), where errors in finding $AB$ led to wrong $x$ and $y$.
Mistake: Assuming AB = BA
Matrix multiplication is not commutative.
Wrong working
$B (A + C) = (A + C) B$ , so multiply in either order.
Right working
Keep the order given: $B (A + C)$ means $B$ on the left.
Do this: Copy the order of letters exactly from the question.
Usually costs: The 2025 Q3(ii) report stresses the order and non-commutative nature of matrix operations.

Exam tips

2024 Q1(iv) and 2025 Q1(v) were both 1-mark MCQs on $A^{2}$ ; a wrong option was always the 'square each entry' matrix, so multiply properly.
In 4-mark questions such as 2024 Q2(i) and 2025 Q3(ii), show each entry's sum (for example $1 \times 4 + 1 \times y$ ); the method mark is for the product.
Before multiplying, write the orders, such as $(2 \times 2) (2 \times 1) = 2 \times 1$ ; it stops you building the wrong-sized answer.
Keep negative entries in brackets: $(- 1) (- 2) = 2$ .

Practice questions on 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 $A$ is of order $1 \times 2$ and $B$ is of order $2 \times 2$ , the order of $A B$ is:
(a) $2 \times 2$
(b) $1 \times 2$
(c) $2 \times 1$
(d) not defined
Hint
Inner numbers match; outer numbers give the order.
Answer
(b) $1 \times 2$
Q2 · 2 marks · Short answerBoard level
Find $[12 - 1 3] [45]$ .
Hint
Each row times the column.
Answer
$[4 - 5 8 + 15] = [- 1 23]$
Q3 · 3 marks · Short answerHarder board level
If $A = [32 - 1 0]$ and $B = [1 - 3 24]$ , find $A B$ and $B A$ .
Hint
Do the two products separately.
Answer
$A B = [6224]$ , $B A = [7 - 1 - 1 3]$ ; $A B \neq = B A$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): If $A = [10 0 - 1]$ , then $A^{2} = I$ .
Reason (R): The square of any matrix is found by squaring each of its entries.
(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
Multiply $A \times A$ ; then test (R) on any other matrix.
Answer
(a). $A \times A = [1001] = I$ , but squaring entries works here only by coincidence; in general it is wrong.
Q5 · 3 marks · Short answerHarder board level
Find $x$ if $[21 x 3] [x 1] = [65]$ .
Hint
The product is $[3 x x + 3]$ .
Answer
$x + 3 = 5$ gives $x = 2$ , and the first row checks: $3 \times 2 = 6$ . So $x = 2$ .
Q6 · 3 marks · Short answerHOTS / competency
Give an example of a $2 \times 2$ matrix $A$ , other than $I$ and $- I$ , for which $A^{2} = I$ . Verify by multiplication.
Hint
Try swapping rows of $I$ .
Answer
$A = [0110]$ : $A^{2} = [0 + 1 0 + 0 0 + 0 1 + 0] = I$ . (Other answers such as $[10 0 - 1]$ are also correct.)
Q7 · 4 marks · Short answerHarder board level
If $A = [1111]$ , show that $A^{2} = 2 A$ and hence that $A^{3} = 4 A$ .
Hint
$A^{3} = A^{2} \cdot A = 2 A \cdot A = 2 A^{2}$ .
Answer
$A^{2} = [2222] = 2 A$ ; $A^{3} = 2 A^{2} = 2 (2 A) = 4 A$ .

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 is the product $A B$ defined?Show answer
When the number of columns of $A$ equals the number of rows of $B$ .
2.What is the order of a $(2 \times 2) (2 \times 1)$ product?Show answer
$2 \times 1$ .
3.Find $[12] [34]$ .Show answer
$[3 + 8] = [11]$ , a $1 \times 1$ matrix.
4.What is $A I$ for any $2 \times 2$ matrix $A$ ?Show answer
$A$ itself.
5.Is matrix multiplication commutative?Show answer
No; in general $A B \neq = B A$ .

Multiplication of Matrices: frequently asked questions

How do you multiply two 2 × 2 matrices?

Each entry of the product is a row of the first matrix times a column of the second: multiply the pairs and add. For example, the top-left entry is (row 1 of A) times (column 1 of B).

Why is A² not found by squaring each element?

Because A² means A × A, and matrix multiplication combines rows with columns. Squaring each element gives a different matrix in almost every case; CISCE examiners reported this error in 2024 and 2025.

Which matrix products does the ICSE syllabus include?

The CISCE syllabus covers compatibility for multiplication and multiplication of a 2 × 2 matrix by a matrix, so you meet 2 × 2 by 2 × 2, 2 × 2 by a column, and a row by 2 × 2.