NCERT Solutions class 12 Maths Exercise 4.5

NCERT Solutions class 12 Maths Exercise 4.5 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

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NCERT Solutions class 12 Maths Determinants

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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NCERT Solutions class 12 Maths Exercise 4.5

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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NCERT Solutions class 12 Maths Exercise 4.5

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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NCERT Solutions class 12 Maths Exercise 4.5

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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NCERT Solutions class 12 Maths Exercise 4.5

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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NCERT Solutions class 12 Maths Exercise 4.5

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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NCERT Solutions class 12 Maths Exercise 4.5

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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NCERT Solutions class 12 Maths Exercise 4.5

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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NCERT Solutions class 12 Maths Exercise 4.5

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.

NCERT Solutions class 12 Maths Exercise 4.5

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