MathsICSE Class 10 · Matrices

Matrix Equations and Equality of Matrices: Finding Unknowns (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

Two matrices are equal only when they have the same order and every pair of corresponding entries is equal. So a matrix equation gives one ordinary equation per entry. Simplify each side to a single matrix first (add, multiply), then equate entries and solve for the unknowns.

Equal matrices match position by position: each same-coloured pair of entries gives one ordinary equation to solve.

Illustration drawn for QuestionBuddy.

Understand Matrix Equations and Equality of Matrices

A matrix equation is solved by turning it into ordinary equations. Everything rests on one rule.

Equality of matrices. $A = B$ only when $A$ and $B$ have the same order and each entry of $A$ equals the entry in the same position of $B$ . For $2 \times 2$ matrices that is four equations:

$[a c b d] = [p r q s] ⟺ a = p, b = q, c = r, d = s .$

Type 1: unknowns inside the entries. Simplify each side to one matrix, then equate. In 2024 Q2(i) the product $A B$ had to be worked out first: $[x 1 01] [4 y 01] = [4 x 4 + y 01]$ . Equating with $[4 x 01]$ gives $4 x = 4$ and $4 + y = x$ , so $x = 1$ , $y = - 3$ . The examiners reported product errors (such as $4 y = 1$ ) as the usual cause of lost marks.

Type 2: an unknown matrix with addition and scalars. Treat it like a linear equation in numbers: $2 X + A = B$ gives $2 X = B - A$ and $X = \frac{1}{2} (B - A)$ . Halve every entry.

**Type 3: a $2 \times 2$ matrix times an unknown column.** $[a c b d] [x y] = [e f]$ means $a x + b y = e$ and $c x + d y = f$ : a pair of simultaneous linear equations. Similarly a row $[x y]$ times a $2 \times 2$ matrix gives two equations. If the unknown is a whole $2 \times 2$ matrix, write it as $[p r q s]$ and you get four equations.

**Type 4: expressions like $A^{2} = p A + q I$ .** Work out $A^{2}$ , write $p A + q I$ as one matrix with $p$ and $q$ inside, and equate entries. You get more equations than unknowns; solve with two and check the rest.

Order first. Before writing an unknown matrix, decide its order from the compatibility rule: if $X$ multiplies a $2 \times 2$ matrix on the left and the answer is $1 \times 2$ , then $X$ must be $1 \times 2$ .

There is no division of matrices in the ICSE syllabus; never write $X = \frac{B}{A}$ .

Matrix Equations and Equality of Matrices: formulas to remember

Equality of matrices: $[a c b d] = [p r q s] ⟺ a = p, b = q, c = r, d = s$; Same order and equal corresponding entries.
Unknown matrix with a scalar: $k X + A = B \Rightarrow X = \frac{1}{k} (B - A)$; $k$ a non-zero number.
Matrix times an unknown column: $[a c b d] [x y] = [e f] ⟺ a x + b y = e, c x + d y = f$; Solve as simultaneous equations.
Combination with the identity: $p A + q I = [p a + q p c p b p d + q] for A = [a c b d]$; $q$ is added only on the leading diagonal.

How to solve Matrix Equations and Equality of Matrices questions

Reduce each side to a single matrix, equate entries, solve, then check.

1Decide the order of any unknown matrix from compatibility, and write it with letters, such as $[x y]$ .
2Simplify each side completely: carry out products, scalar multiples and sums.
3Equate corresponding entries; write each equation on its own line and number them.
4Solve the simplest equation first; use simultaneous equations if needed.
5Substitute your values back into the original matrices to check every entry, including those you did not use.

Matrix Equations and Equality 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.

Matrix equation as simultaneous equations

Suppose a cinema's two evening shows sold these tickets: show 1, 40 adult and 30 child tickets, taking ₹13,000; show 2, 50 adult and 20 child tickets, taking ₹14,500.

Write this as a matrix equation and find the price of each kind of ticket.

Try it first, then show the solutionSolution

$[40503020] [a c] = [1300014500]$ , where $a$ and $c$ are the adult and child prices.
Equating entries: $40 a + 30 c = 13000$ and $50 a + 20 c = 14500$ , that is $4 a + 3 c = 1300$ and $5 a + 2 c = 1450$ .
Multiply by 2 and 3: $8 a + 6 c = 2600$ and $15 a + 6 c = 4350$ ; subtract: $7 a = 1750$ , $a = 250$ ; then $3 c = 300$ , $c = 100$ .

Answer: Adult ticket ₹250, child ticket ₹100.

Memory hook: One row of the matrix equation = one sentence of the story.

Unknown matrix from addition

Imagine two branches of a stationery shop that want to start next week with 50 pens and 50 files each. Their current stock is $S = [30251240]$ (rows: branches; columns: pens, files).

Find the delivery matrix $X$ so that $S + X$ is the target stock.

Try it first, then show the solutionSolution

Target $T = [50505050]$ ; $S + X = T$ gives $X = T - S$ .
$X = [50 - 30 50 - 25 50 - 12 50 - 40] = [20253810]$ .

Answer: Branch 1 needs 20 pens and 38 files; branch 2 needs 25 pens and 10 files.

Memory hook: Move a matrix across the equals sign exactly as you would a number.

Tricks to remember Matrix Equations and Equality of Matrices

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

Four boxes, four equations
Equal $2 \times 2$ matrices give four equations, one from each box; a column gives two.
Use it when: Knowing how many equations you have before solving.
Simplify, then equate
Never equate entries until each side is a single matrix.
Use it when: Questions like $A B = C$ or $2 X + A = B$ .
I lives on the diagonal
$q I$ adds $q$ only to the top-left and bottom-right entries.
Use it when: Equations such as $A^{2} = p A + q I$ .

Matrix Equations and Equality of Matrices: solved examples

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

Example 1: Equal matrices · 1 mark · MCQBoard level

If $[x + y 1 2 x - y] = [6122]$ , then $(x, y)$ is:
(a) $(4, 2)$
(b) $(2, 4)$
(c) $(3, 3)$
(d) $(5, 1)$

Try it first, then show the solutionSolution

Equating entries: $x + y = 6$ and $x - y = 2$ .
Adding: $2 x = 8$ , $x = 4$ ; then $y = 2$ .

Answer: (a) $(4, 2)$

Example 2: Unknown matrix with a scalar · 3 marks · Short answerBoard level

Find the matrix $X$ if $X + [23 - 1 0] = 2 [1024]$ .

Try it first, then show the solutionSolution

Right side: $2 [1024] = [2048]$ .
$X = [2048] - [23 - 1 0] = [0 - 3 58]$ .

Answer: $X = [0 - 3 58]$

Example 3: Unknown column: simultaneous equations · 4 marks · Short answerHarder board level

Find $x$ and $y$ if $[31 - 2 4] [x y] = [8 - 2]$ .

Try it first, then show the solutionSolution

Multiplying: $[3 x - 2 y x + 4 y] = [8 - 2]$ .
So $3 x - 2 y = 8$ … (1) and $x + 4 y = - 2$ … (2).
From (2), $x = - 2 - 4 y$ . In (1): $- 6 - 12 y - 2 y = 8$ , so $- 14 y = 14$ , $y = - 1$ .
Then $x = - 2 + 4 = 2$ . Check (1): $6 + 2 = 8$ .

Answer: $x = 2$ , $y = - 1$

Example 4: Unknowns from AB = C · 4 marks · Short answerHarder board level

$A = [20 x 1]$ , $B = [1 y 03]$ and $C = [- 2 - 2 63]$ . Find $x$ and $y$ if $A B = C$ .

Try it first, then show the solutionSolution

$A B = [2 \times 1 + x \times y 0 \times 1 + 1 \times y 2 \times 0 + x \times 3 0 \times 0 + 1 \times 3] = [2 + x y y 3 x 3]$ .
Equating with $C$ : $3 x = 6$ gives $x = 2$ ; $y = - 2$ .
Check the remaining entry: $2 + x y = 2 + 2 (- 2) = - 2$ ✓.

Answer: $x = 2$ , $y = - 2$

Example 5: Finding p and q in A² = pA + qI · 4 marks · Short answerHOTS / competency

If $A = [42 - 1 1]$ , find $p$ and $q$ such that $A^{2} = p A + q I$ .

Try it first, then show the solutionSolution

$A^{2} = [16 - 2 8 + 2 - 4 - 1 - 2 + 1] = [1410 - 5 - 1]$ .
$p A + q I = [4 p + q 2 p - p p + q]$ .
Top-right: $- p = - 5$ , so $p = 5$ . Top-left: $20 + q = 14$ , so $q = - 6$ .
Check the others: $2 p = 10$ ✓ and $p + q = - 1$ ✓.

Answer: $p = 5$ , $q = - 6$ (so $A^{2} - 5 A + 6 I = O$ ).

Example 6: Case study: prices from two bills · 4 marks · Case-basedHOTS / competency

At a fruit stall, Ravi buys 3 kg apples and 2 kg bananas for ₹100; Meera buys 1 kg apples and 4 kg bananas for ₹90. Let the prices be ₹ $a$ and ₹ $b$ per kg.
(a) Write the information as a matrix equation.
(b) Find $a$ and $b$ .
(c) What would 5 kg of each cost?

Try it first, then show the solutionSolution

(a) $[3124] [a b] = [10090]$ .
(b) $3 a + 2 b = 100$ and $a + 4 b = 90$ . From the second, $a = 90 - 4 b$ ; then $270 - 12 b + 2 b = 100$ , so $b = 17$ and $a = 22$ .
(c) $5 \times 22 + 5 \times 17 = 110 + 85 = 195$ .

Answer: (b) Apples ₹22 per kg, bananas ₹17 per kg (c) ₹195

Where students lose marks in Matrix Equations and Equality of Matrices

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

Mistake: Equating before multiplying correctly
Each entry of $A B$ is a row-times-column sum; a wrong product gives wrong equations.
Wrong working
$A B$ has bottom-left entry $4 y$ , so $4 y = 1$ and $y = \frac{1}{4}$ .
Right working
Bottom-left entry is $1 \times 4 + 1 \times y = 4 + y$ ; $4 + y = x = 1$ , so $y = - 3$ .
Do this: Write each entry's sum in full before equating.
Usually costs: Reported in 2024 Q2(i); most of the 4 marks.
Mistake: Adding q to every entry of pA + qI
$q I$ has $q$ only on the leading diagonal and 0 elsewhere.
Wrong working
$p A + q I = [4 p + q 2 p + q - p + q p + q]$
Right working
$p A + q I = [4 p + q 2 p - p p + q]$
Do this: Write $q I = [q 0 0 q]$ as a separate step.
Usually costs: The final answer and accuracy marks.
Mistake: Halving only part of 2X
Scalar division acts on every entry.
Wrong working
$2 X = [4226]$ , $X = [2223]$
Right working
$X = [2113]$
Do this: Divide all four entries and check by doubling.
Usually costs: One accuracy mark.

Exam tips

2024 Q2(i) (4 marks) asked for $x$ and $y$ given $A B = C$ ; the expected working was the full product $A B$ , then one equation per entry.
Use the simplest equation first (often a single-letter one), then substitute into the harder one.
After solving, check the entries you did not use; if one fails, recheck your product.
Write the conclusion clearly, for example ' $x = 1$ , $y = - 3$ '.

Practice questions on Matrix Equations and Equality 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 $[2 a 4 3 b + 1] = [84 3 - 2]$ , then:
(a) $a = 4, b = - 3$
(b) $a = 4, b = - 1$
(c) $a = 16, b = - 3$
(d) $a = 2, b = - 3$
Hint
$2 a = 8$ and $b + 1 = - 2$ .
Answer
(a) $a = 4$ , $b = - 3$
Q2 · 3 marks · Short answerBoard level
Solve for $X$ : $2 X - [1 - 2 35] = [34 - 1 1]$ .
Hint
Add the matrix to both sides, then halve.
Answer
$2 X = [4226]$ , so $X = [2113]$ .
Q3 · 3 marks · Short answerHarder board level
Find $x$ and $y$ if $[13 2 - 1] [x y] = [70]$ .
Hint
$x + 2 y = 7$ and $3 x - y = 0$ .
Answer
$x = 1$ , $y = 3$
Q4 · 3 marks · Short answerHarder board level
Find $x$ and $y$ if $[x y] [13 2 - 1] = [94]$ .
Hint
The product is $[x + 3 y 2 x - y]$ .
Answer
$x + 3 y = 9$ and $2 x - y = 4$ give $x = 3$ , $y = 2$ .
Q5 · 3 marks · Short answerHarder board level
If $A = [a 1 20]$ and $A^{2} = [11362]$ , find $a$ .
Hint
$A^{2} = [a^{2} + 2 a 2 a 2]$ .
Answer
$a = 3$ (from $2 a = 6$ ; check $a^{2} + 2 = 11$ ).
Q6 · 4 marks · Short answerHarder board level
Find the $2 \times 2$ matrix $X$ such that $[1201] X = [3815]$ .
Hint
Let $X = [p r q s]$ ; the product is $[p 2 p + r q 2 q + s]$ .
Answer
$p = 3$ , $q = 1$ , $r = 2$ , $s = 3$ : $X = [3213]$ .
Q7 · 4 marks · Short answerHOTS / competency
Find $x$ and $y$ if $[x - y y x] [12] = [50]$ . Hence find $M^{2}$ where $M = [x - y y x]$ .
Hint
$x + 2 y = 5$ and $2 x - y = 0$ .
Answer
$x = 1$ , $y = 2$ ; $M^{2} = [1 - 2 21]^{2} = [- 3 - 4 4 - 3]$ .

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 are two matrices equal?Show answer
When they have the same order and all corresponding entries are equal.
2.Find $x$ if $[x - 1 0] = [40]$ .Show answer
$x = 5$ .
3.Solve $2 X = [4062]$ .Show answer
$X = [2031]$ .
4.How many equations does $[a c b d] [x y] = [e f]$ give?Show answer
Two: $a x + b y = e$ and $c x + d y = f$ .
5.What is $3 I$ ?Show answer
$[3003]$ .

Matrix Equations and Equality of Matrices: frequently asked questions

How do you solve a matrix equation in ICSE Class 10?

Simplify each side to a single matrix, then use equality of matrices: corresponding entries are equal. This gives ordinary equations, which you solve by substitution or elimination.

Can I divide one matrix by another?

No. Division of matrices is not defined in the ICSE syllabus. Instead, write the unknown matrix with letters and equate entries.

What order should the unknown matrix X have?

Use compatibility. If X times a 2 × 2 matrix gives a 1 × 2 matrix, X must be 1 × 2; if a 2 × 2 matrix times X gives a 2 × 1 matrix, X must be 2 × 1.