# NCERT Solutions for Exercise 4.2 Class 12 Maths Chapter 4 - Determinants

In this article, you will get NCERT solutions for Class 12 Maths chapter 4 exercise 4.2. These Exercise 4.2 Class 12 Maths solutions are consist of questions related to properties of determinants. Properties of determinants make it easy for us to finding determinants without complicated calculations. There are 6 properties of determinants related to operation on the determinants given in the NCERT textbook before the Class 12 Maths ch 4 ex 4.2. You are advised to go through the proof of these properties given in the textbook to get a better understanding. There are some examples given after each property which will also help you to get conceptual clarity. Here you can get NCERT Solutions by clicking on this link.

**Also, see**

- Determinants Exercise 4.1
- Determinants Exercise 4.3
- Determinants Exercise 4.4
- Determinants Exercise 4.5
- Determinants Exercise 4.6
- Determinants Miscellaneous Exercise

**Determinants Exercise:4.2**

**Question:1** Using the property of determinants and without expanding, prove that

**Answer:**

We can split it in manner like;

So, we know the identity that If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero.

Clearly, expanded determinants have identical columns.

**Hence the sum is zero.**

**Question:2 **Using the property of determinants and without expanding, prove that

**Answer:**

Given determinant

Applying the rows addition then we have;

So, we have two rows and identical hence we can say that the value of determinant = 0

Therefore .

**Question:3 **Using the property of determinants and without expanding, prove that

**Answer:**

Given determinant

So, we can split it in two addition determinants:

[ Here two columns are identical ]

and [ Here two columns are identical ]

**Therefore we have the value of determinant = 0. **

**Question:4 **Using the property of determinants and without expanding, prove that

**Answer:**

We have determinant:

Applying we have then;

**So, here column 3 and column 1 are proportional.**

**Therefore, .**

**Question:5 **Using the property of determinants and without expanding, prove that

**Answer:**

Given determinant :

Splitting the third row; we get,

.

Then we have,

On Applying row transformation and then ;

we get,

Applying Rows exchange transformation and , we have:

also

On applying rows transformation, and then

and then

Then applying rows exchange transformation;

and then . we have then;

So, we now calculate the sum =

**Hence proved.**

**Question:6** Using the property of determinants and without expanding, prove that

**Answer:**

We have given determinant

Applying transformation, we have then,

We can make the first row identical to the third row so,

Taking another row transformation: we have,

So, determinant has two rows identical.

**Hence .**

**Question:7 **Using the property of determinants and without expanding, prove that

**Answer:**

Given determinant :

As we can easily take out the common factors **a,b,c** from rows respectively.

So, get then:

Now, taking common factors **a,b,c** from the columns respectively.

Now, applying rows transformations and then we have;

**Expanding to get R.H.S.**

**Question:8(i) **By using properties of determinants, show that:

**Answer:**

We have the determinant

Applying the row transformations and then we have:

Now, applying we have:

or

**Hence proved.**

**Question:8(ii)** By using properties of determinants, show that:

**Answer:**

Given determinant :

,

Applying column transformation and then

We get,

Now, applying column transformation , we have:

**Hence proved.**

**Question:9 **By using properties of determinants, show that:

**Answer:**

We have the determinant:

Applying the row transformations and then , we have;

Now, applying ; we have

Now, expanding the remaining determinant;

**Hence proved.**

**Question:10(i) **By using properties of determinants, show that:

**Answer:**

Given determinant:

Applying row transformation: then we have;

Taking a common factor: **5x+4**

**Now, applying column transformations and **

**Question:10(ii) **By using properties of determinants, show that:

**Answer:**

Given determinant:

Applying row transformation we get;

[taking common (3y + k) factor]

Now, applying column transformation and

**Hence proved.**

**Question:11(i) **By using properties of determinants, show that:

**Answer:**

Given determinant:

We apply row transformation: we have;

**Taking common factor (a+b+c) out.**

Now, applying column tranformation and then

We have;

**Hence Proved.**

**Question:11(ii)** By using properties of determinants, show that:

**Answer:**

Given determinant

Applying we get;

Taking **2(x+y+z)** factor out, we get;

Now, applying row transformations, and then .

we get;

**Hence proved.**

**Question:12** By using properties of determinants, show that:

**Answer:**

Give determinant

Applying column transformation we get;

[**after taking the (1+x+x ^{2 }) factor common out.]**

Now, applying row transformations, and then .

we have now,

As we know

**Hence proved.**

**Question:13 **By using properties of determinants, show that:

**Answer:**

We have determinant:

Applying row transformations, and then we have;

taking common factor out of the determinant;

Now expanding the remaining determinant we get;

**Hence proved.**

**Question:14** By using properties of determinants, show that:

**Answer:**

Given determinant:

Let

Then we can clearly see that each column can be reduced by taking common factors like **a,b,** and** c **respectively from** C _{1,}C_{2,}**and

**C**

_{3.}We then get;

Now, applying column transformations: and

then we have;

Now, expanding the remaining determinant:

.

**Hence proved.**

**Question:15** Choose the correct answer. Let A be a square matrix of order , then is equal to

(A) (B) (C) (D)

**Answer:**

Assume a square matrix A of order of .

Then we have;

(**Taking the common factors k from each row.**)

**Therefore correct option is (C).**

**Question:16** Choose the correct answer.

Which of the following is correct

(A) Determinant is a square matrix.

(B) Determinant is a number associated to a matrix.

(C) Determinant is a number associated to a square matrix.

(D) None of these

**Answer:**

**The answer is (C) **Determinant is a number associated to a square matrix.

As we know that To every square matrix of order n, we can associate a number (real or complex) called determinant of the square matrix A, where element of A.

**More About NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2**

This article NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2 is consists of questions related to properties of determinants. In Class 12th Maths chapter 4 exercise 4.2 there are 16 questions including 2 multiple choice type questions. There are 11 examples given in NCERT book before the exercise 4.2 Class 12 Maths. First, try to solve these examples given in the textbook. It will help you to get conceptual clarity and solving NCERT problems. NCERT syllabus **Class 12th Maths chapter 4 exercise 4.2** questions are very important for the board exam as well as for the engineering competitive exams.

**Also Read**| Determinants Class 12 Chapter 4 Notes

**Benefits of ****NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2**

- Class 12 Maths chapter 4 exercise 4.2 solutions are prepared by the subject matter experts who know how best to answer in order to perform well in the board exams.
- Class 12th Maths chapter 4 exercise 4.2 questions are prepared in a very descriptive manner which you will get easily.
- NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2 are important in competitive exams like JEE, SRMJEE, etc.
- As most of the time, one question from this exercise is asked in the board exam, so you are advised to be thorough with them.
- You can use these NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2 for reference.

**Also see-**

NCERT Solutions for Class 12 Maths Chapter 4

NCERT Exemplar Solutions Class 12 Maths Chapter 4

**NCERT Solutions of Class 12 Subject Wise**

NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Physics

NCERT Solutions for Class 12 Chemistry

NCERT Solutions for Class 12 Biology

**Subject Wise NCERT Exampler Solutions**

NCERT Exemplar Solutions for Class 12th Maths

NCERT Exemplar Solutions for Class 12th Physics

NCERT Exemplar Solutions for Class 12th Chemistry

NCERT Exemplar Solutions for Class 12th Biology

**Happy learning!!!**