Class 12 NCERT Solutions- Mathematics Part I - Chapter 4 Determinants - Exercise 4.2 | Set 1

Exercise 4.2 specifically focuses on the properties and applications of determinants, building upon the basic definitions introduced earlier in the chapter. This exercise aims to deepen students' understanding of how determinants can be manipulated and applied to solve various mathematical problems.

In this section, students are introduced to more advanced techniques for evaluating 3x3 determinants, moving beyond the elementary calculations covered in previous exercises. The properties of determinants, such as the relationship between a determinant and its transpose, are explored in depth. These properties not only simplify calculations but also provide insights into the underlying structure of determinants.

Using the properties of determinants and without expanding in Exercise 1 to 7, prove that:

Question 1. 

Solution:

L.H.S.=

                C₁→C₁+C₂

=

According to Properties of Determinant   

=0                                [∵  C₁ & C₃ are identical]

Now, L.H.S.=R.H.S.

Hence Proved

Question 2. 

Solution:

L.H.S.=

=0                        [∵ Every element of C₁ are 0]

Now,  L.H.S.=R.H.S.

Hence Proved

Question 3. 

Solution:

L.H.S.=

            C₃→C₃-C₁ 

        =

      =

=9 ×0=0          [∵C₂ & C₃  are identical]

Now,  L.H.S.=R.H.S.

Hence Proved

Question 4. 

Solution:

L.H.S.=

        =

Now,  L.H.S.=R.H.S.

Hence Proved

Question 5. 

Solution:

L.H.S.=

Now, L.H.S.=R.H.S.

Hence Proved

Question 6. 

Solution:

Let Δ=

Taking (-1) common from every row

  Δ=(-1)³

Interchange rows and columns

Δ=-

Now, Δ=-Δ

       Δ+Δ=0

       2Δ=0

        Δ=0

Now,  L.H.S.=R.H.S.

Hence Proved

Question 7. 

Solution:

L.H.S.=

Taking common a from Row 1,

                          b from Row 2,

                          c from Row 3, we have

Now,  L.H.S.=R.H.S.

Hence Proved

By using properties of determinants, in Exercises 8 to 14, show that:

Question 8(i). 

(ii)

Solution:

(i) L.H.S.=

Now,  L.H.S.=R.H.S.

Hence Proved

(ii)  L.H.S.=

Now,  L.H.S.=R.H.S.

Hence Proved

Question 9. 

Solution:

L.H.S.=

Now, L.H.S.=R.H.S.

Hence Proved

Question 10.(i)

(ii)

Solution:

(i) L.H.S.=

Now,  L.H.S.=R.H.S.

Hence Proved

(ii) L.H.S.=

Now,  L.H.S.=R.H.S.

Hence Proved

Summary

Exercise 4.2 of Chapter 4 serves as a comprehensive exploration of determinants, their properties, and applications. Through a series of carefully crafted problems, students gain proficiency in evaluating and manipulating 3x3 determinants, applying key properties to simplify calculations and prove identities.The exercise reinforces several important concepts, including the equivalence of a determinant and its transpose, the effect of row and column operations on determinants, and the relationship between a matrix's determinant and its scalar multiples. Students learn to recognize special determinant forms that either equal zero or have simplified expressions, enhancing their ability to work efficiently with these structures.