NCERT Solutions Class 12

Exercise 4.2

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.

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2. 

Ans. On  Operating 

= 

=  = 0 = R.H.S.

[ All entries of one column here first are zero]

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3.  

Ans. On  operating 

=  = 

= 9 x 0 = 0 [ two columns are identical] Proved.

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4.  

Ans. Given:      = 

Operating  

= 

=  = 0 [ two columns are identical] Proved.

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5.  

Ans. L.H.S. =  operating 

= 

= 

= 

=  [operating ]

=  [operating ]

=  [operating ]

=  [Interchanging  and ]

=  [Interchanging  and ]

=  = R.H.S.

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6. 

Ans. Let 

=  [Taking  common from each row]

Interchanging rows and columns in the determinants on R.H.S.,

 

 

  Proved.

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7.  

Ans. L.H.S. = 

Taking common  from  respectively,

= 

=    [operating ]

=  = 

=  =  = R.H.S.

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8. (i)  

(ii)  

Ans. (i) L.H.S. =  operating  and ,

= 

=  [Expanding along 1^st column]

= 

= 

=  =  = R.H.S. Proved.

(ii) L.H.S. =  operating  and 

= 

= 

= 

= 

=  

= 

= 

= 

= 

=  = R.H.S.

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9.  

Ans. 

L.H.S. =  =  [Multiplying  by  respectively]

=  = 

=  [operating ]

= 

= 

= 

= 

= 

= 

= 

= 

= 

= 

=  = R.H.S.

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10. (i)  

(ii)  

Ans. (i) L.H.S. = 

=  

= 

=  [operating  and ]

= 

=  = R.H.S.

(ii) L.H.S. = 

=  

= 

=  [operating  and ]

= 

=  = 

= R.H.S.  Proved.

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11. (i)  

(ii)  

Ans. (i) L.H.S. = 

=  

= 

=  

= 

= 

=  = R.H.S. Proved.

(ii) L.H.S. = 

=  

= 

=  

= 

= 

=  = R.H.S. Proved.

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12. 

Ans. L.H.S. = 

=  

= 

=  

= 

= 

= 

= 

= 

= 

= 

=  = R.H.S.      Proved.

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13. 

Ans. L.H.S. = 

=  

= 

=  

= 

= 

=  = R.H.S.

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14.  

Ans. L.H.S. = 

Multiplying  by  respectively and then dividing the determinant by 

= 

= 

=  

= 

=  

= 

=  = R.H.S.        Proved.

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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.

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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.