NCERT Solutions Class 12

Miscellaneous

1. Let A =  show that  where I is the identity matrix of order 2 and  N.

Ans. Using Mathematical Induction, we see the result is true for  for

Given:  is true, i.e. 

To prove: 

Proof: L.H.S.  =  =  = 

= 

= 

=  = R.H.S.

Thus,  is true, therefore,  is true.

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2. If A = , prove that A’’ =  N.

Ans. Given: A =           ……….(i)

Let  

    = 

   is true for 

Now   ……….(ii)

Multiplying eq. (ii) by eq. (i),  

  = 

Therefore,  is true for all natural numbers by P.M.I.

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3. If A =  then prove that A’’ =  where  is any positive integer.

Ans. Given: A’’ =       

   which is true for 

Now,         ……….(i)

Again        ……….(ii)

    [From eq. (i)]

   = 

  

Therefore, the result is true for 

Hence, by the principal of mathematical induction, the result is true for all positive integers 

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4. If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.

Ans. A and B are symmetric matrices.    A’ = A and B’ = B  ……….(i)

Now, (AB – BA)’ = (AB)’ – (BA)’    (AB – BA)’ = B’A’ – A’B’ [Reversal law]

 (AB – BA)’ = BA – AB   [Using eq. (i)]

 (AB – BA)’ = – (AB – BA)

Therefore, (AB – BA) is a skew symmetric.

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5. Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Ans. (B’AB)’ = [B’(AB]’ = (AB)’ (B’)’   [ (CD)’ = D’C’]

  (B’AB)’ = B’A’B   ……….(i)

Case I: A is a symmetric matrix, then   A’ = A

  From eq. (i)  (B’AB)’ = B’AB

 B’AB is a symmetric matrix.

Case II: A is a skew symmetric matrix.  A’ = – A

Putting A’ = – A in eq. (i),  (B’AB)’ = B’(– A)B = – B’AB

 B’AB is a skew symmetric matrix.

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6. Find the values of  if the matrix A =  satisfies the equation A’A = I.

Ans. Given: A = 

  A’ = 

Now A’A = I   

 

 

Equating corresponding entries, we have

       

And       

And       

  , , 

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7. For what value of   ?

Ans. Given: 

  

       

Equating corresponding entries, we have

       

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8. If A =  show that A² – 5A + 7I = 0.

Ans. Given: A = 

  A² – 5A + 7I = 

=  = 

=  =  = 

=  = 0 = R.H.S.        Proved.

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9. Find  if  

Ans. Given: 

  

  

 

 

 

Equating corresponding entries, we have

      

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10. A manufacturer produces three products,  which he sells in two markets. Annual sales are indicated below:

	Market	Products
I. 10,000	2,000	18,000
II. 6,000	20,000	8,000

 (a) If unit sales prices of  and  are ` 2.50, ` 1.50 and ` 1.00 respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.

Ans. According to question, the matrix  A = 

(a) Let B be the column matrix representing sale price of each unit of products 

Then B = 

Now Revenue = Sale price c Number of items sold

   =  = 

Therefore, the revenue collected by sale of all items in Market I = ` 46,000 and the revenue collected by sale of all items in Market II = ` 53,000.

(b) Let C be the column matrix representing cost price of each unit of products 

Then C = 

  Total cost = AC = 

=  = 

  The profit collected in two markets is given in matrix form as

Profit matrix = Revenue matrix – Cost matrix

 

Therefore, the gross profit in both the markets = ` 15000 + ` 17000 = ` 32,000.

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11. Find the matrix X so that X  

Ans. Given: X  ……….(i)

Putting X =  in eq. (i),  

  

Equating corresponding entries, we have

   …..(ii)      …..(iii)

 …..(iv)      …..(v)

…..(vi)      …..(vi)

Solving eq. (ii) and (iii), we have    and 

Solving eq. (v) and (vi), we have    and 

Putting these values in X = ,    X = 

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12. If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB’’ = B’’A. Further prove that (AB)’’ = A’’B’’ for all  N.

Ans. Given: AB = BA …..(i)

Let       ……(ii)

For     becomes AB = BA

   is true for 

For      

Multiplying both sides by B,      

     [From eq. (i)]

   is also true.

Therefore,  is true for all  N by P.M.I.

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13. If A =  is such that A² = I, then:

(A)   

(B)   

(C)   

(D)  

Ans. Given: A =  and A² = I

  

  

  

Equating corresponding entries, we have

 

Therefore, option (C) is correct.

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14. If the matrix A is both symmetric and skew symmetric, then:

(A) A is a diagonal matrix  

(B) A is a zero matrix

(C) A is a square matrix  

(D) None of these

Ans. Since, A is symmetric, therefore, A’ = A ……..(i)

And A is skew-symmetric, therefore, A’ = – A

  A = – A  [From eq. (i)]

  A + A = 0  2A = 0   A = 0

Therefore, A is zero matrix.

Therefore, option (B) is correct.

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15. If A is a square matrix such that A² = A, then (I + A)³ – 7A is equal to:

(A) A  

(B) I – A  

(C) I  

(D) 3A

Ans. Given: A² = A    …..(i)

Multiplying both sides by A,  A³ = A² = A [From eq. (i)]  ……(ii)

Also given (I + A)³ – 7A = I³ + A³ + 3I²A + 3IA² – 7A

Putting A² = A [from eq. (i)] and A³ = A [from eq. (ii)],

= I + A + 3IA + 3IA – 7A = I + A + 3A + 3A – 7A   [ IA = A]

= I + 7A – 7A = I

Therefore, option (C) is correct.