Short Intro

Chapter 3 of Class 12 Mathematics (Matrices) focuses on the foundational concepts of arrays. Exercise 3.1 is designed to clear up core ideas regarding the order, elements, construction, and equality of matrices. This comprehensive guide provides step-by-step solutions for all 10 problems in the exercise to help students ace their board and competitive examinations.

Quick Information Box

Key Metric	Details
Board / Syllabus	NCERT / CBSE / State Boards (Class 12)
Chapter Name	Chapter 3: Matrices
Exercise	Exercise 3.1
Total Questions	10 (Short answer, long answer, and MCQs)
Core Topics	Order of Matrix, Element Identification, Matrix Construction, Equality of Matrices

Concepts Used (Topics Covered)

To solve this exercise seamlessly, you need to understand the following primary definitions:

1. Definition of Matrix: An ordered rectangular array of numbers or functions.
2. Order of a Matrix: A matrix having m rows and n columns is said to be of order m×n.
3. Number of Elements: An m×n matrix contains exactly mn elements.
4. Construction of Matrices: Creating matrix structures using a specified element-generation formula aij​.
5. Equality of Matrices: Two matrices are equal if they share the exact same order and their corresponding elements are identical.

Important Formulas

1. Matrix representation: A=[a_ij​]_m×n​
2. Total number of elements = Rows (m)×Columns (n)=mn
3. Condition for Equality: If A=[a_ij​] and B=[b_ij​] are equal, then:

   • Order of A=Order of B
   • a_ij​=b_ij​ (for all i,j)

4. Condition for Square Matrix: Number of rows = Number of columns (m=n).

Questions & Step-by-Step Solutions

Question 1

Question 2

Question 3

Question 4

Construct a 2×2 matrix, A=[a_ij​], whose elements are given by:

(i)

(ii)

(iii)

Question 5

Construct a 3×4 matrix, whose elements are given by:

(i)

(ii)

Question 6

Find the values of x, y and z from the following equations:

(i)

(ii)

(iii)

Question 7

Find the value of a, b, c and d from the equation:

Question 8

Question 9

Question 10

Common Mistakes

Students often understand the basic definition of a matrix but lose marks because of small errors involving indices, signs, fractions, or corresponding positions. The following mistakes should be avoided.

1. Writing the order in reverse

For a matrix,$$\text{Order} = \text{Number of rows} \times \text{Number of columns}$$

For example, a matrix having 3 rows and 4 columns is a $3\times4$ matrix, not a $4\times3$ matrix.

A simple memory rule is:$$\boxed{\text{Rows first, Columns second}}$$

2. Confusing a_ij with a_ji

The element $a_{ij}$means:$$a_{ij}=\text{element in the }i^{th}\text{ row and }j^{th}\text{ column}$$

Therefore,$$a_{13}\neq a_{31}$$

in general.

For example, if$$A= \begin{bmatrix} 2&5&19&-7\\ 35&-2&\frac52&12\\ \sqrt3&1&-5&17 \end{bmatrix},$$

then$$a_{13}=19$$

whereas$$a_{31}=\sqrt3.$$

3. Forgetting that the number of elements is mn

For an $m\times n$ matrix:$$\boxed{\text{Number of elements}=mn}$$

For example:$$3\times4=12\text{ elements}$$

and$$3\times3=9\text{ elements}.$$

4. Missing reverse factor pairs

If a matrix has 24 elements, students sometimes write only:$$1\times24,\quad 2\times12,\quad 3\times8,\quad 4\times6$$

and forget:$$24\times1,\quad 12\times2,\quad 8\times3,\quad 6\times4.$$

Because matrix order is an ordered pair,$$m\times n\neq n\times m$$

unless $m=n$

5. Using the wrong range of i and j

For a $3\times4$ matrix:$$i=1,2,3$$

and$$j=1,2,3,4.$$

A common mistake is to use $i=1,2,3,4$ and $j=1,2,3$.

6. Ignoring modulus signs

In Question 5(i),$$a_{ij}=\frac12|-3i+j|$$

The modulus is essential. A negative value inside the modulus becomes positive.

For example,$$a_{21} = \frac12|-3(2)+1| = \frac12|-5| = \frac52.$$

Do not write $-\frac52$−.

7. Comparing the wrong elements in equal matrices

If$$A=B,$$

then only elements in exactly the same positions are compared.

For example,$$\begin{bmatrix} x+y&2\\ 5+z&xy \end{bmatrix} = \begin{bmatrix} 6&2\\ 5&8 \end{bmatrix}$$

gives:$$x+y=6,\qquad 5+z=5,\qquad xy=8.$$

Do not compare $x+y$ with 2 or $xy$ with 5.

8. Solving only some of the equations

In equality-of-matrices questions, all corresponding equations must be satisfied simultaneously.

For example, in Question 9:$$3x+7=0$$

gives$$x=-\frac73.$$

Also,$$y-2=5$$

gives$$y=7.$$

But then:$$y+1=8$$

is satisfied, while$$2-3x=4$$

would require$$x=-\frac23.$$

Since the values of $x$x conflict, the matrices cannot be equal. Therefore, the correct option is (B) Not possible to find.

9. Using addition instead of multiplication principle

For a $3\times3$ matrix whose entries can be either 0 or 1:$$9\text{ positions}$$

and each position has:$$2\text{ choices}.$$

Therefore:$$2^9=512,$$

not $2\times9=18$

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Exam Tips

For Exercise 3.1, the following strategies can improve speed and accuracy.

Tip 1: Draw a small row-column locator

Whenever a question asks for $a_{ij}$ mentally read:$$\boxed{a_{ij}\rightarrow \text{Row }i,\text{ Column }j}$$

For example:$$a_{24}\rightarrow \text{2nd row, 4th column}.$$

Tip 2: Use factor pairs for order questions

If the number of elements is $N$, find all positive integer pairs satisfying:$$mn=N.$$

For 18:$$18=1\times18=2\times9=3\times6.$$

Then include reverse orders:$$18\times1,\quad 9\times2,\quad 6\times3.$$

Tip 3: Make an i-j substitution table

For constructing a matrix from $a_{ij}$, prepare a table:

Element	$i$i	$j$j
$a_{11}$	1	1
$a_{12}$	1	2
$a_{21}$	2	1
$a_{22}$	2	2

For larger matrices, calculate row by row.

Tip 4: Check the final matrix order

After constructing a $3\times4$ matrix, count:

• 3 horizontal rows
• 4 entries in each row

This quick check prevents incomplete matrices.

Tip 5: In matrix equality, write equations position-wise

Use the pattern:$$R_1C_1,\quad R_1C_2,\quad R_2C_1,\quad R_2C_2.$$

This prevents skipping equations.

Tip 6: Remember the square matrix condition

For$$A=[a_{ij}]_{m\times n},$$

the matrix is square only when:$$\boxed{m=n}.$$

Tip 7: Use kmn for counting matrices

If a matrix has order $m\times n$, it contains $mn$ positions.

If every entry has $k$ choices, then:$$\boxed{\text{Total matrices}=k^{mn}}.$$

For entries 0 or 1:$$k=2.$$

Thus, for a $3\times3$ matrix:$$2^{3\times3}=2^9=512.$$

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Practice MCQs

MCQ 1

A matrix has 4 rows and 7 columns. Its order is:

(A) $7\times4$
(B) $4\times7$
(C) $28\times1$
(D) $11\times1$

Answer: (B) $4\times7$

Explanation: Matrix order is written as rows × columns.

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MCQ 2

How many elements are present in a $5\times3$ matrix?

(A) 8
(B) 15
(C) 53
(D) 2

Answer: (B) 15$$5\times3=15.$$

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MCQ 3

If a matrix contains 17 elements, which of the following can be its order?

(A) $17\times1$
(B) $1\times17$
(C) Both A and B
(D) $2\times8$

Answer: (C) Both A and B

Since 17 is prime, its only factor pair is:$$17=1\times17.$$

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MCQ 4

In the matrix$$A= \begin{bmatrix} 4&7&9\\ 2&5&8 \end{bmatrix},$$

the value of $a_{23}$​ is:

(A) 2
(B) 5
(C) 8
(D) 9

Answer: (C) 8

The element in row 2 and column 3 is 8.

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MCQ 5

A matrix $A=[a_{ij}]_{m\times n}$is square when:

(A) $m<n$
(B) $m>n$
(C) $m=n$
(D) $m+n=0$

Answer: (C) $m=n$

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MCQ 6

If$$a_{ij}=i+j,$$

then $a_{23}$ is:

(A) 5
(B) 6
(C) 1
(D) 23

Answer: (A) 5$$a_{23}=2+3=5.$$

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MCQ 7

If$$a_{ij}=2i-j,$$

then $a_{32}$​ is:

(A) 1
(B) 4
(C) 8
(D) -4

Answer: (B) 4$$a_{32}=2(3)-2=4.$$

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MCQ 8

If$$\begin{bmatrix} x&3\\ 4&y \end{bmatrix} = \begin{bmatrix} 2&3\\ 4&7 \end{bmatrix},$$

then:

(A) $x=2,\ y=7$
(B) $x=7,\ y=2$
(C) $x=3,\ y=4$
(D) None

Answer: (A) $x=2,\ y=7$

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MCQ 9

The number of possible $2\times2$ matrices whose entries are either 0 or 1 is:

(A) 4
(B) 8
(C) 16
(D) 32

Answer: (C) 16

A $2\times2$matrix has 4 entries:$$2^4=16.$$

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MCQ 10

The number of possible $2\times3$ matrices with each entry selected from $\{0,1,2\}$ is:

(A) 18
(B) 64
(C) 243
(D) 729

Answer: (D) 729

There are:$$2\times3=6$$

positions and 3 choices per position.

Therefore:$$3^6=729.$$

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FAQ Section

1. What is a matrix?

A matrix is a rectangular arrangement of numbers or mathematical expressions organised into rows and columns.

For example:$$A= \begin{bmatrix} 1&2&3\\ 4&5&6 \end{bmatrix}$$

is a matrix of order $2\times3$.

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2. How is the order of a matrix determined?

The order is:$$\boxed{\text{Number of rows}\times\text{Number of columns}}$$

If a matrix has 3 rows and 4 columns, its order is:$$3\times4.$$

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3. How many elements are there in an $m\times n$ matrix?

The number of elements is:$$\boxed{mn}.$$

For example, a $4\times5$ matrix has:$$4\times5=20$$

elements.

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4. What does $a_{ij}$​ represent?

The notation $a_{ij}$ represents the element in:$$i^{th}\text{ row and }j^{th}\text{ column}.$$

Thus,$$a_{32}$$

is the element in the third row and second column.

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5. Can matrices of different orders be equal?

No. For two matrices to be equal:

1. Their orders must be the same.
2. Their corresponding elements must be equal.

Both conditions are compulsory.

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6. What are the possible orders of a matrix having 24 elements?

Since:$$mn=24,$$

the possible orders are:$$1\times24,\quad 2\times12,\quad 3\times8,\quad 4\times6,$$

and their reverse orders:$$24\times1,\quad 12\times2,\quad 8\times3,\quad 6\times4.$$

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7. Why does a matrix with 13 elements have only two possible orders?

Because 13 is a prime number.

Its only positive factorisation is:$$13=1\times13.$$

Therefore, the possible orders are:$$1\times13$$

and$$13\times1.$$

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8. What is a square matrix?

A matrix with an equal number of rows and columns is called a square matrix.

Thus:$$A=[a_{ij}]_{m\times n}$$

is square when:$$\boxed{m=n}.$$

Examples include:$$2\times2,\quad 3\times3,\quad 4\times4.$$

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9. How do we construct a matrix when $a_{ij}$​ is given?

Follow these steps:

1. Write the general form of the required matrix.
2. Identify the possible values of $i$ and $j$.
3. Substitute each $(i,j)$ pair into the given formula.
4. Calculate every element carefully.
5. Arrange the results in their correct row-column positions.

For example, for a $2\times2$ matrix:$$A= \begin{bmatrix} a_{11}&a_{12}\\ a_{21}&a_{22} \end{bmatrix}.$$

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10. Why is Question 9 answered as “Not possible to find”?

The matrices in Question 9 produce inconsistent requirements for $x$x.

From the first corresponding elements:$$3x+7=0$$

so:$$x=-\frac73.$$

However, from the bottom-right corresponding elements:$$2-3x=4$$

which gives:$$x=-\frac23.$$

The same variable cannot simultaneously have both values. Hence, no values of $x$x and $y$y can make the two matrices equal.

Therefore:$$\boxed{\text{Option (B): Not possible to find}}$$Option (B): Not possible to find​

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11. How many $3\times3$ matrices can be formed using only 0 and 1?

A $3\times3$ matrix contains:$$3\times3=9$$

entries.

Each entry has two choices:$$0\text{ or }1.$$

Therefore:$$2^9=512.$$

Hence:$$\boxed{512}$$​

different matrices can be formed.

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12. What is the most important concept in Exercise 3.1?

The most important concepts are:

• order of a matrix,
• number and location of elements,
• notation $a_{ij}$aij​,
• construction of matrices using formulas,
• equality of matrices,
• square matrix condition, and
• multiplication principle for counting matrices.

Students who are comfortable with these concepts will find later topics such as matrix addition, multiplication, transpose, and inverse easier to understand.

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CTA – Start Practising with MyMockMate

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