Exercise 4.2
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NCERT Solutions class 12 Maths Determinants
Using the properties of determinants and without expanding in Exercise 1 to 5, prove that:
1. 
Ans. Given:
Operating 
0 = 0 [
C1 and C2 are identical]
L.H.S. = R.H.S.
2. 
Ans. On 
Operating 
= 
= 
= 0 = R.H.S.
[ All entries of one column here first are zero]
3. 
Ans. On 
operating 
= 
= 
= 9 x 0 = 0 [ two columns are identical] Proved.
4. 
Ans. Given: 
= 
Operating 
= 
= 
= 0 [ two columns are identical] Proved.
5. 
Ans. L.H.S. = 
operating 
= 
= 
= 
= 
[operating 
]
= 
[operating 
]
= 
[operating 
]
= 
[Interchanging 
and 
]
= 
[Interchanging and ]
= 
= R.H.S.
6. 
Ans. Let 
= 
[Taking 
common from each row]
Interchanging rows and columns in the determinants on R.H.S.,
Proved.
7. 
Ans. L.H.S. = 
Taking common 
from 
respectively,
= 
= 
[operating 
]
= 
= 
= 
= 
= R.H.S.
8. (i) 
(ii) 
Ans. (i) L.H.S. = 
operating 
and ,
= 
= 
[Expanding along 1st column]
= 
= 
= 
= 
= R.H.S. Proved.
(ii) L.H.S. = 
operating 
and
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= R.H.S.
9. 
Ans.
L.H.S. = 
= 
[Multiplying 
by 
respectively]
= 
= 
= 
[operating 
]
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= R.H.S.
10. (i) 
(ii) 
Ans. (i) L.H.S. = 
= 
= 
= 
[operating and ]
= 
= 
= R.H.S.
(ii) L.H.S. = 
= 
= 
= 
[operating and ]
= 
= 
= 
= R.H.S. Proved.
11. (i) 
(ii) 
Ans. (i) L.H.S. = 
= 
= 
= 
= 
= 
= 
= R.H.S. Proved.
(ii) L.H.S. = 
= 
= 
= 
= 
= 
= 
= R.H.S. Proved.
12. 
Ans. L.H.S. = 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= R.H.S. Proved.
13. 
Ans. L.H.S. = 
= 
= 
= 
= 
= 
= 
= R.H.S.
14. 
Ans. L.H.S. = 
Multiplying 
by respectively and then dividing the determinant by 
= 
= 
= 
= 
= 
= 
= 
= R.H.S. Proved.
NCERT Solutions class 12 Maths Exercise 4.2
Choose the correct answer in Exercises 15 and 16.
15. Let A be a square matrix of order 3 x 3, then 
is equal to:
(A) 
(B) 
(C) 
(D) 
Ans. Let A = 
be a square matrix of order 3 x 3. ……….(i)
= [From eq. (i)]
Therefore, option (C) is correct.
NCERT Solutions class 12 Maths Exercise 4.2
16. Which is 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.
Ans. Since, Determinant is a number associated to a square matrix.
Therefore, option (C) is correct.