NCERT Solutions Class 12

Exercise 4.5

Find adjoint of each of the matrices in Exercise 1 and 2.

1.  

Ans. Here A = 

 

 A₁₁ = Cofactor of 

A₁₂ = Cofactor of 

A₂₁ = Cofactor of 

A₂₂ = Cofactor of 

 adj. A =  = 

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2.  

Ans. Here A = 

 

      = 

 

 

 

 adj. A = 

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Verify A (adj. A) =  in Exercise 3 and 4.

3.  

Ans. Let A = 

 adj. A = 

 A.(adj. A) = 

=  =   …..(i)

Again  (adj. A). A = 

=  =     …..(ii)

And  = 

Again      …..(iii)

 From eq. (i), (ii) and (iii)  A. (adj. A) = (adj. A). A = 

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4.  

Ans. Let A = 

   

    = 

  

  

   

 adj. A = 

 A. (adj. A) = 

= 

=  ……….(i)

Again  (adj. A). A = 

= 

=   ……….(ii)

And 

= 

Also  =     ……….(iii)

 From eq. (i), (ii) and (iii)  A. (adj. A) = (adj. A). A = 

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Find the inverse of the matrix (if it exists) given in Exercise 5 to 11.

5.  

Ans. Let A = 

  =  0

 Matrix A is non-singular and hence  exist.

Now adj. A =  And 

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6.  

Ans. Let A = 

  = 

 Matrix A is non-singular and hence  exist.

Now adj. A =  And 

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7.  

Ans. Let A = 

  = 

  exists.

A₁₁ = ,  A₁₂ = ,

A₁₃ = ,   A₂₁ = ,

A₂₂ = ,   A₂₃ = ,

A₃₁ = ,   A₃₂ = ,

A₃₃ = 

 adj. A = 

 

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8.  

Ans. Let A = 

   = 

  exists.

A₁₁ = , A₁₂ = ,

A₁₃ = , A₂₁ = ,

A₂₂ = , A₂₃ = ,

A₃₁ = ,   A₃₂ = ,

A₃₃ = 

 adj. A = 

 

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9.  

Ans. Let A = 

  = 

  exists.

A₁₁ = , A₁₂ = ,

A₁₃ = , A₂₁ = ,

A₂₂ = , A₂₃ = ,

A₃₁ = ,  A₃₂ = ,

A₃₃ = 

 adj. A = 

 

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10.  

Ans. Let A = 

  = 

  exists.

A₁₁ = ,  A₁₂ = ,

A₁₃ = , A₂₁ = ,

A₂₂ = , A₂₃ = ,

A₃₁ = , A₃₂ = ,

A₃₃ = 

 adj. A = 

 

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11.  

Ans. Let A = 

 

= 

  exists.

A₁₁ = ,

A₁₂ = , A₁₃ = ,

A₂₁ = ,   A₂₂ = ,

A₂₃ = ,  A₃₁ = ,

A₃₂ = , A₃₃ = 

 adj. A = 

 

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12. Let A =  and B =  verify that  

Ans. Given: Matrix A = 

  = 15 – 14 = 1  0

  = 

Matrix B = 

  = 54 – 56 =   0

 

Now AB =  =  = 

  = 

Now L.H.S. =     ……….(i)

R.H.S. = 

= 

=   ……….(ii)

 From eq. (i) and (ii), we get

L.H.S. = R.H.S.

 

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13. If A = , show that A² – 5A + 7I = 0. Hence find  

Ans. Given: A = 

  

 

L.H.S. = 

= 

= 

= 

= 

= 

= R.H.S.

  ……(i)

To find: , multiplying eq. (i) by .

 

 

 

= 

=  = 

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14. For the matrix A =  find numbers  and  such that  

Ans. Given: A = 

  

 

 

 

 

 

 We have  ……….(i)

 

 

Here  satisfies  also, therefore  

Putting  in eq. (i),        

Here also  satisfies  , therefore    

Therefore,  and 

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15. For the matrix A = , show that  Hence find 

Ans. Given: A = 

 

  = 

Now 

= 

= 

L.H.S. = 

= 

= 

= 

=  =  = R.H.S.

Now,   to find , multiplying  by 

 

 

 

 

 

  = 

 

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16. If A = , verify that  and hence find 

Ans. Given: A = 

 

  = 

Now 

= 

= 

L.H.S. = 

= 

= 

= 

=  =  = R.H.S.

Now,   to find , multiplying  by 

 

 

 

 

 

  = 

 

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17. Let A be a non-singular matrix of order 3 x 3. Then  is equal to:

(A) 

(B)    

(C)    

(D)  

Ans. If A is a non-singular matrix of order  then 

 Putting  

Therefore, option (B) is correct.

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18. If A is an invertible matrix of order 2, then det  is equal to:

(A) det A 

(B)    

(C) 1  

(D) 0

Ans. Since 

  

 

 

Therefore, option (B) is correct.