## NCERT Solutions for Class 12 Maths Exercise Miscellaneous NCERT Solutions for Class 12 Maths Chapter 3 â€“ Matrices â€“ FREE PDF Download

NCERT Class 12 Maths Ch 3 is one of the most important ones in the NCERT syllabus. Duly following NCERT Solutions for Class 12 Maths Chapter 3 ensures you that there will be no hindrance when you opt for more advanced branches of Maths. Class 12 Maths Chapter 3 â€“ Matrices solved by Expert Teachers as per NCERT (CBSE) Book guidelines. All Matrices Exercise Questions with Solutions to help you to revise complete Syllabus and Score More marks.

# NCERT Solutions for Class 12 Maths Chapter 3 â€“ Matrices

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

Given:Â Â is true, i.e.Â

To prove:Â

Proof:Â L.H.S.Â =Â Â =Â Â =Â

=Â

=Â

=Â Â = R.H.S.

Thus,Â Â is true, therefore,Â Â is true.

### 2. If A =Â , prove that Aâ€™â€™ =Â Â N.

**Ans.**Â Given: A =Â Â Â Â Â Â Â Â Â Â Â â€¦â€¦â€¦.(i)

LetÂ Â

Â Â Â Â =Â

Â Â Â is true forÂ

NowÂ Â Â â€¦â€¦â€¦.(ii)

Multiplying eq. (ii) by eq. (i),Â Â

Â Â =Â

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

### 3. If A =Â Â then prove that Aâ€™â€™ =Â Â whereÂ Â is any positive integer.

**Ans.**Â Given: Aâ€™â€™ =Â Â Â Â Â Â Â

Â Â Â which is true forÂ

Now,Â Â Â Â Â Â Â Â Â â€¦â€¦â€¦.(i)

AgainÂ Â Â Â Â Â Â Â â€¦â€¦â€¦.(ii)

Â Â Â Â [From eq. (i)]

Â Â Â =Â

Â Â

Therefore, the result is true forÂ

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

### 4. If A and B are symmetric matrices, prove that AB â€“ BA is a skew symmetric matrix.

**Ans.**Â A and B are symmetric matrices.Â Â Â Â Aâ€™ = A and Bâ€™ = BÂ â€¦â€¦â€¦.(i)

Now, (AB â€“ BA)â€™ = (AB)â€™ â€“ (BA)â€™Â Â Â Â (AB â€“ BA)â€™ = Bâ€™Aâ€™ â€“ Aâ€™Bâ€™ [Reversal law]

Â (AB â€“ BA)â€™ = BA â€“ ABÂ Â [Using eq. (i)]

Â (AB â€“ BA)â€™ = â€“ (AB â€“ BA)

Therefore, (AB â€“ BA) is a skew symmetric.

**Ans.**Â (Bâ€™AB)â€™ = [Bâ€™(AB]â€™ = (AB)â€™ (Bâ€™)â€™Â Â [ Â (CD)â€™ = Dâ€™Câ€™]

Â (Bâ€™AB)â€™ = Bâ€™Aâ€™BÂ Â â€¦â€¦â€¦.(i)

Case I: A is a symmetric matrix, thenÂ Â Â Aâ€™ = A

Â From eq. (i)Â (Bâ€™AB)â€™ = Bâ€™AB

Â Bâ€™AB is a symmetric matrix.

Case II: A is a skew symmetric matrix.Â Â Aâ€™ = â€“ A

Putting Aâ€™ = â€“ A in eq. (i),Â (Bâ€™AB)â€™ = Bâ€™(â€“ A)B = â€“ Bâ€™AB

Â Bâ€™AB is a skew symmetric matrix.

### 6. Find the values ofÂ Â if the matrix A =Â Â satisfies the equation Aâ€™A = I.

**Ans.**Â Given: A =Â

Â Â Aâ€™ =Â

Now Aâ€™A = IÂ Â Â

Â

Â

Equating corresponding entries, we have

Â Â Â Â Â Â Â

AndÂ Â Â Â Â Â Â

AndÂ Â Â Â Â Â Â

Â Â ,Â ,Â

### 7. For what value ofÂ Â Â ?

**Ans.**Â Given:Â

Â Â

Â Â Â Â Â Â Â

Equating corresponding entries, we have

Â Â Â Â Â Â Â

### 8. If A =Â
Â show that A^{2}Â â€“ 5A + 7I = 0.

**Ans.**Â Given: A =Â

Â A

^{2}Â â€“ 5A + 7I =Â

=Â Â =Â

=Â Â =Â Â =Â

=Â Â = 0 = R.H.S.Â Â Â Â Â Â Â Proved.

### 9. FindÂ Â ifÂ Â

**Ans.**Â Given:Â

Â Â

Â Â

Â

Â

Â

Equating corresponding entries, we have

Â Â Â Â Â Â

### 10. A manufacturer produces three products,Â Â 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 ofÂ Â andÂ Â 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.**Â According to question, the matrix Â A =Â

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

Then B =Â

Now Revenue = Sale price c Number of items sold

Â Â Â =Â Â =Â

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)Â Let C be the column matrix representing cost price of each unit of productsÂ

Then C =Â

Â Total cost = AC =Â

=Â Â =Â

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

Profit matrix = Revenue matrix â€“ Cost matrix

Â

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

**Ans.**Â Given: X Â Â â€¦â€¦â€¦.(i)

Putting X =Â Â in eq. (i),Â Â

Â Â

Equating corresponding entries, we have

Â Â â€¦..(ii)Â Â Â Â Â Â â€¦..(iii)

Â â€¦..(iv)Â Â Â Â Â Â â€¦..(v)

â€¦..(vi)Â Â Â Â Â Â â€¦..(vi)

Solving eq. (ii) and (iii), we haveÂ Â Â Â andÂ

Solving eq. (v) and (vi), we haveÂ Â Â Â andÂ

Putting these values in X =Â ,Â Â Â X =Â

12. If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABâ€™â€™ = Bâ€™â€™A. Further prove that (AB)â€™â€™ = Aâ€™â€™Bâ€™â€™ for allÂ Â N.

**Ans.**Â Given: AB = BA â€¦..(i)

LetÂ Â Â Â Â Â Â â€¦â€¦(ii)

ForÂ Â Â Â Â becomes AB = BA

Â Â Â is true forÂ

ForÂ Â Â Â Â Â

Multiplying both sides by B,Â Â Â Â Â Â

Â Â Â Â Â [From eq. (i)]

Â Â Â is also true.

Therefore,Â Â is true for allÂ Â N by P.M.I.

^{2}Â = I, then:

(A)Â

(B)Â

(C)Â

(D)Â

**Ans.**Â Given: A =Â Â and A

^{2}Â = I

Â Â

Â Â

Â Â

Equating corresponding entries, we have

Â

Therefore, option (C) is correct.

### 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.**Â Since, A is symmetric, therefore, Aâ€™ = A â€¦â€¦..(i)

And A is skew-symmetric, therefore, Aâ€™ = â€“ A

Â A = â€“ AÂ [From eq. (i)]

Â A + A = 0Â Â 2A = 0Â Â Â A = 0

Therefore, A is zero matrix.

Therefore, option (B) is correct.

### 15. If A is a square matrix such that A^{2}Â = A, then (I + A)^{3}Â â€“ 7A is equal to:

(A) A

(B) I â€“ A

(C) I

(D) 3A

**Ans.**Â Given: A

^{2}Â = AÂ Â Â â€¦..(i)

Multiplying both sides by A,Â A

^{3}Â = A

^{2}Â = A [From eq. (i)]Â â€¦â€¦(ii)

Also given (I + A)

^{3}Â â€“ 7A = I

^{3}Â + A

^{3}Â + 3I

^{2}A + 3IA

^{2}Â â€“ 7A

Putting A

^{2}Â = A [from eq. (i)] and A

^{3}Â = A [from eq. (ii)],

= I + A + 3IA + 3IA â€“ 7A = I + A + 3A + 3A â€“ 7AÂ Â [ Â IA = A]

= I + 7A â€“ 7A = I

Therefore, option (C) is correct.