Question 1

If a matrix has $3$ rows and $2$ columns, its order is:

Accepted Answer

**Answer: (a) $3	imes2$.** Order is written as (number of rows) $	imes$ (number of columns) $=3	imes2$.

Question 2

If $A$ is of order $2	imes3$ and $B$ is of order $3	imes4$, then the order of $AB$ is:

Accepted Answer

**Answer: (a) $2	imes4$.** The inner dimensions match ($3=3$), so $AB$ is defined with order (rows of $A$) $	imes$ (columns of $B$) $=2	imes4$.

Question 3

If $A=\begin{pmatrix}2&3\1&0\end{pmatrix}$, then $2A=$

Accepted Answer

**Answer: (a) $\begin{pmatrix}4&6\2&0\end{pmatrix}$.** Scalar multiplication multiplies every element by $2$: $\begin{pmatrix}2(2)&2(3)\2(1)&2(0)\end{pmatrix}=\begin{pmatrix}4&6\2&0\end{pmatrix}$.

Question 4

If $A=\begin{pmatrix}1&2\3&4\end{pmatrix}$, then $A^2=$

Accepted Answer

**Answer: (a) $\begin{pmatrix}7&10\15&22\end{pmatrix}$.** $A^2=A\cdot A=\begin{pmatrix}1+6&2+8\3+12&6+16\end{pmatrix}=\begin{pmatrix}7&10\15&22\end{pmatrix}$.

Question 5

**Assertion (A):** Matrix multiplication is not commutative. **Reason (R):** For two matrices $A$ and $B$, the products $AB$ and $BA$ may be unequal even when both are defined.

Accepted Answer

**Answer: (a)** Non-commutativity means precisely that $AB
eq BA$ in general, which is what R states, so R is the correct explanation of A.

Question 6

If $A=\begin{pmatrix}3&1\2&4\end{pmatrix}$ and $B=\begin{pmatrix}1&2\0&5\end{pmatrix}$, find $A-B$.

Accepted Answer

Subtract corresponding elements: $$A-B=\begin{pmatrix}3-1&1-2\2-0&4-5\end{pmatrix}=\begin{pmatrix}2&-1\2&-1\end{pmatrix}.$$

Question 7

If $\begin{pmatrix}x+y\x-y\end{pmatrix}=\begin{pmatrix}6\2\end{pmatrix}$, find the values of $x$ and $y$.

Accepted Answer

Equating corresponding elements: $x+y=6$ and $x-y=2$. Adding: $2x=8\Rightarrow x=\mathbf{4}$; then $y=6-4=\mathbf{2}$.

Question 8

If $A=\begin{pmatrix}2&1\3&2\end{pmatrix}$ and $B=\begin{pmatrix}1&0\-1&2\end{pmatrix}$, find the product $AB$.

Accepted Answer

$$AB=\begin{pmatrix}2&1\3&2\end{pmatrix}\begin{pmatrix}1&0\-1&2\end{pmatrix}.$$ Row 1: $(2\cdot1+1\cdot(-1),\ 2\cdot0+1\cdot2)=(1,2)$. Row 2: $(3\cdot1+2\cdot(-1),\ 3\cdot0+2\cdot2)=(1,4)$. $$AB=\begin{pmatrix}1&2\1&4\end{pmatrix}.$$

Question 9

Find the matrix $X$ such that $2X+\begin{pmatrix}1&2\3&4\end{pmatrix}=\begin{pmatrix}5&6\7&8\end{pmatrix}$.

Accepted Answer

$2X=\begin{pmatrix}5&6\7&8\end{pmatrix}-\begin{pmatrix}1&2\3&4\end{pmatrix}=\begin{pmatrix}4&4\4&4\end{pmatrix}.$ Divide every element by $2$: $$X=\begin{pmatrix}2&2\2&2\end{pmatrix}.$$

Question 10

If $A=\begin{pmatrix}3&x\0&1\end{pmatrix}$ and $A^2=\begin{pmatrix}9&12\0&1\end{pmatrix}$, find the value of $x$.

Accepted Answer

$$A^2=\begin{pmatrix}3&x\0&1\end{pmatrix}\begin{pmatrix}3&x\0&1\end{pmatrix}=\begin{pmatrix}9&3x+x\0&1\end{pmatrix}=\begin{pmatrix}9&4x\0&1\end{pmatrix}.$$ Comparing with $\begin{pmatrix}9&12\0&1\end{pmatrix}$: $4x=12\Rightarrow x=\mathbf{3}$.

Question 11

If $A=\begin{pmatrix}1&2\3&4\end{pmatrix}$ and $I$ is the $2	imes2$ identity matrix, show that $A^2-5A-2I=O$, where $O$ is the zero matrix.

Accepted Answer

**Compute $A^2$:** $$A^2=\begin{pmatrix}1&2\3&4\end{pmatrix}\begin{pmatrix}1&2\3&4\end{pmatrix}=\begin{pmatrix}7&10\15&22\end{pmatrix}.$$ **Compute $5A$ and $2I$:** $$5A=\begin{pmatrix}5&10\15&20\end{pmatrix},\qquad 2I=\begin{pmatrix}2&0\0&2\end{pmatrix}.$$ **Combine:** $$A^2-5A-2I=\begin{pmatrix}7-5-2&10-10-0\15-15-0&22-20-2\end{pmatrix}=\begin{pmatrix}0&0\0&0\end{pmatrix}=O.$$ Hence $A^2-5A-2I=O$.

Question 12

Given $A=\begin{pmatrix}3&-2\4&-2\end{pmatrix}$ and $I=\begin{pmatrix}1&0\0&1\end{pmatrix}$, find the value of $k$ such that $A^2=kA-2I$.

Accepted Answer

**Compute $A^2$:** $$A^2=\begin{pmatrix}3&-2\4&-2\end{pmatrix}\begin{pmatrix}3&-2\4&-2\end{pmatrix}=\begin{pmatrix}9-8&-6+4\12-8&-8+4\end{pmatrix}=\begin{pmatrix}1&-2\4&-4\end{pmatrix}.$$ **Write $kA-2I$:** $$kA-2I=\begin{pmatrix}3k-2&-2k\4k&-2k-2\end{pmatrix}.$$ **Equate corresponding elements** (top-left): $3k-2=1\Rightarrow k=1$. Check with the other entries: $-2k=-2$, $4k=4$, $-2k-2=-4$ — all hold for $k=1$. Hence $k=\mathbf{1}$.

Question 13

In a stationery shop, the quantities of pens and notebooks bought by two students are given by the matrix $Q=\begin{pmatrix}2&3\\4&1\end{pmatrix}$ (rows = students, columns = pens and notebooks). The prices in rupees are given by the column matrix $P=\begin{pmatrix}5\\20\end{pmatrix}$ (pen ₹ $5$ , notebook ₹ $20$ ).

(i) State the order of the product $QP$ .
(ii) Compute $QP$ and interpret its entries.
(iii) Which student spends more, and by how much?

Accepted Answer

**(i)** $Q$ is $2	imes2$ and $P$ is $2	imes1$, so $QP$ has order $\mathbf{2	imes1}$. **(ii)** $$QP=\begin{pmatrix}2&3\4&1\end{pmatrix}\begin{pmatrix}5\20\end{pmatrix}=\begin{pmatrix}2(5)+3(20)\4(5)+1(20)\end{pmatrix}=\begin{pmatrix}70\40\end{pmatrix}.$$ Student 1 spends ₹$70$ and Student 2 spends ₹$40$. **(iii)** Student 1 spends more, by $70-40=$ ₹$\mathbf{30}$.

Question 14

Are these Matrices important questions free?

Accepted Answer

Yes. All 13 ICSE Class 10 Maths important questions for Matrices are free, with full model answers and no login required.

Question 15

Do these Matrices questions follow the latest ICSE syllabus?

Accepted Answer

Yes — they are aligned to the CISCE latest syllabus syllabus for ICSE Class 10 Maths, so nothing here is outside the current course.

Question 16

How should I practise the Matrices important questions?

Accepted Answer

Attempt each question on paper first, then reveal the model answer to check your method — not just the final result. Re-do anything you got wrong the same day.

Question 17

What types of questions are covered for Matrices?

Accepted Answer

A full mix — multiple-choice questions, assertion–reason questions, very short answer questions, short answer questions, long answer questions, case-based questions — so every format in the ICSE paper is covered.

Question type	Count	Marks
MCQ	4	1
Assertion–Reason	1	1
Very Short	2	2
Short Answer	3	3
Long Answer	2	5
Case-based	1	4

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