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NCERT Solutions Class 12 maths Chapter-3 (matrices)Exercise 3.3

NCERT Solutions Class 12 maths Chapter-3 (matrices)Exercise 3.3

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-3 (matrices)Exercise 3.3 This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.
Solutions Class 12 maths Chapter-3 (matrices)Exercise 3.3

Exercise 3.3

Question 1. Find the transpose of each of the following matrices:

(i) Solutions Class 12 maths Chapter-3 (matrices) 

(ii) Solutions Class 12 maths Chapter-3 (matrices)

(iii) Solutions Class 12 maths Chapter-3 (matrices)

Solution:


(i) Let A =Solutions Class 12 maths Chapter-3 (matrices)

∴Transpose of A = A’ = A = Solutions Class 12 maths Chapter-3 (matrices)

(ii) Let A =Solutions Class 12 maths Chapter-3 (matrices)

∴Transpose of A = A’ = AT  =Solutions Class 12 maths Chapter-3 (matrices)

(iii) Let A =Solutions Class 12 maths Chapter-3 (matrices)

∴Transpose of A = A’ = AT  =Solutions Class 12 maths Chapter-3 (matrices)

Question 2. If A =Solutions Class 12 maths Chapter-3 (matrices) and B = Solutions Class 12 maths Chapter-3 (matrices)then verify that:

(i) (A+B)’ = A’+B’

(ii) (A-B)’ = A’- B’

Solution:


(i) A+B =Solutions Class 12 maths Chapter-3 (matrices)

L.H.S. = (A+B)’ = Solutions Class 12 maths Chapter-3 (matrices)

R.H.S. = A’+B’ = Solutions Class 12 maths Chapter-3 (matrices)

∴L.H.S = R.H.S.

Hence, proved.

(ii) A-B = Solutions Class 12 maths Chapter-3 (matrices)

L.H.S. = (A-B)’Solutions Class 12 maths Chapter-3 (matrices)

R.H.S. = A’-B’ =Solutions Class 12 maths Chapter-3 (matrices)

∴ L.H.S. = R.H.S.

Hence, proved.

Question 3.  If A’ =Solutions Class 12 maths Chapter-3 (matrices) and B = Solutions Class 12 maths Chapter-3 (matrices)

, then verify that:

(i) (A+B)’=A’+B’

(ii) (A-B)’=A’-B’

Solution:


Given A’=Solutions Class 12 maths Chapter-3 (matrices)and B=Solutions Class 12 maths Chapter-3 (matrices)

then, (A’)’ = A =Solutions Class 12 maths Chapter-3 (matrices)

(i) A+B =Solutions Class 12 maths Chapter-3 (matrices)

∴ L.H.S. =  (A+B)’=Solutions Class 12 maths Chapter-3 (matrices)

R.H.S.= A’+B’ = Solutions Class 12 maths Chapter-3 (matrices)

∴ L.H.S. = R.H.S.

Hence, proved.

(ii) A-B = Solutions Class 12 maths Chapter-3 (matrices)

∴ L.H.S. =  (A-B)’=Solutions Class 12 maths Chapter-3 (matrices)

R.H.S.= A’-B’ = Solutions Class 12 maths Chapter-3 (matrices)

∴ L.H.S. = R.H.S.

Hence, proved.

Question 4. If A’Solutions Class 12 maths Chapter-3 (matrices)and B = Solutions Class 12 maths Chapter-3 (matrices)

then find (A+2B)’.

Solution:


Given: A’ =Solutions Class 12 maths Chapter-3 (matrices)and B =Solutions Class 12 maths Chapter-3 (matrices)

then (A’)’ =A=Solutions Class 12 maths Chapter-3 (matrices)

Now, A+2B = Solutions Class 12 maths Chapter-3 (matrices)

∴(A+2B)’ = Solutions Class 12 maths Chapter-3 (matrices)

Question 5. For the matrices A and B, verify that (AB)′ = B′A′, where

(i) A =Solutions Class 12 maths Chapter-3 (matrices) and B = Solutions Class 12 maths Chapter-3 (matrices)

(ii) A =Solutions Class 12 maths Chapter-3 (matrices) and B =Solutions Class 12 maths Chapter-3 (matrices)

Solution:


(i) AB = =Solutions Class 12 maths Chapter-3 (matrices)

∴  L.H.S. = (AB)′ =Solutions Class 12 maths Chapter-3 (matrices)

R.H.S.= B′A’ = Solutions Class 12 maths Chapter-3 (matrices)

∴ L.H.S. = R.H.S.

Hence, proved.

(ii) AB =Solutions Class 12 maths Chapter-3 (matrices)

∴  L.H.S. = (AB)′ =Solutions Class 12 maths Chapter-3 (matrices)

Now, R.H.S.=B’A’ = Solutions Class 12 maths Chapter-3 (matrices)

∴ L.H.S. = R.H.S.

Hence, proved.

Question 6. If (i) A =Solutions Class 12 maths Chapter-3 (matrices)  , then verify that A′ A = I.

(ii) A =Solutions Class 12 maths Chapter-3 (matrices)

,then verify that A′ A = I.

Solution:


(i) Solutions Class 12 maths Chapter-3 (matrices)

= I = R.H.S.

∴ L.H.S. = R.H.S.

(ii) Solutions Class 12 maths Chapter-3 (matrices)

= I = R.H.S.

∴ L.H.S. = R.H.S.

Question 7. (i) Show that the matrix ASolutions Class 12 maths Chapter-3 (matrices) =is a symmetric matrix.

(ii) Show that the matrix ASolutions Class 12 maths Chapter-3 (matrices) =is a symmetric matrix.

Solution:

(i) Given: A =Solutions Class 12 maths Chapter-3 (matrices)     

Now, A’=Solutions Class 12 maths Chapter-3 (matrices) 

∵ A = A’

∴ A is a symmetric matrix.

(ii) Given: A = Solutions Class 12 maths Chapter-3 (matrices)

Now, A’=Solutions Class 12 maths Chapter-3 (matrices)

∵ A = A’

∴ A is a symmetric matrix.

Question 8.  For the matrix =Solutions Class 12 maths Chapter-3 (matrices),, verify that:

(i) (A + A′) is a symmetric matrix

(ii) (A – A′) is a skew symmetric matrix

Solution:


(i) Given: A =Solutions Class 12 maths Chapter-3 (matrices)

Let B = (A+A’) = Solutions Class 12 maths Chapter-3 (matrices)

Now, B’ = (A+A’)’ = Solutions Class 12 maths Chapter-3 (matrices)

∵ B = B’

∴ B=(A+A’) is a symmetric matrix.

(ii) Given: A =Solutions Class 12 maths Chapter-3 (matrices)

Let B = (A-A’) =Solutions Class 12 maths Chapter-3 (matrices)

Now, B’ = (A-A’)’ =Solutions Class 12 maths Chapter-3 (matrices)

∵ -B = B’

∴ B=(A-A’) is a skew symmetric matrix.

Question 9. Find 1/2(A+A’) and 1/2(A-A’) ,when A =Solutions Class 12 maths Chapter-3 (matrices).

Solution:


Given: A = Solutions Class 12 maths Chapter-3 (matrices)

∴  A’ = Solutions Class 12 maths Chapter-3 (matrices)

Now,  A+A’ = +Solutions Class 12 maths Chapter-3 (matrices)

Now, A-A’ =Solutions Class 12 maths Chapter-3 (matrices)

Question 10. Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

(i) Solutions Class 12 maths Chapter-3 (matrices)

(ii) Solutions Class 12 maths Chapter-3 (matrices)

(iii) Solutions Class 12 maths Chapter-3 (matrices)

(iv) Solutions Class 12 maths Chapter-3 (matrices)

Solution:


(i) Given : A =Solutions Class 12 maths Chapter-3 (matrices)

⇒ A’=Solutions Class 12 maths Chapter-3 (matrices)

Let P = Solutions Class 12 maths Chapter-3 (matrices)

and Q = Solutions Class 12 maths Chapter-3 (matrices)

Now, P =Solutions Class 12 maths Chapter-3 (matrices)…..(1)

& P’ = Solutions Class 12 maths Chapter-3 (matrices)

∵ P=P’

∴ P is a symmetric matrix.

Now, Q =Solutions Class 12 maths Chapter-3 (matrices)…..(2)

& Q’ = Solutions Class 12 maths Chapter-3 (matrices)

∵ -Q=Q’ 

∴ Q is a skew symmetric matrix.

By adding (1) and (2), we get,

Solutions Class 12 maths Chapter-3 (matrices)

Therefore, A =P + Q

(ii) Given : Solutions Class 12 maths Chapter-3 (matrices)

⇒ A’=Solutions Class 12 maths Chapter-3 (matrices)

P = Solutions Class 12 maths Chapter-3 (matrices)

…..(1)

Q = Solutions Class 12 maths Chapter-3 (matrices)

……(2)

By adding (1) and (2), we get,

Solutions Class 12 maths Chapter-3 (matrices)

Therefore, A =P + Q

(iii) Given: A =Solutions Class 12 maths Chapter-3 (matrices)

⇒ A’=Solutions Class 12 maths Chapter-3 (matrices)

P = }Solutions Class 12 maths Chapter-3 (matrices)…..(1)

Q = Solutions Class 12 maths Chapter-3 (matrices)……(2)

By adding (1) and (2), we get

}Solutions Class 12 maths Chapter-3 (matrices)

Therefore, A =P + Q

(iv) Given: A = Solutions Class 12 maths Chapter-3 (matrices)

⇒ A’= Solutions Class 12 maths Chapter-3 (matrices)

P =Solutions Class 12 maths Chapter-3 (matrices)

…..(1)

Q = Solutions Class 12 maths Chapter-3 (matrices)

…..(2)

By adding (1) and (2), we get

Solutions Class 12 maths Chapter-3 (matrices)

Therefore, A =P + Q

Question 11. If A, B are symmetric matrices of same order, then AB – BA is a

(A) Skew symmetric matrix (B) Symmetric matrix

(C) Zero matrix (D) Identity matrix 

Solution:


Given: A and B are symmetric matrices.

⇒ A=A’

⇒ B=B’

Now, ( AB – BA)’ =(AB)’-(BA)’              [∵ (X-Y)’=X’-Y’]

                          =B’A’-A’B’                [∵ (XY)’=Y’X’]

                         =BA-AB                   [∵ Given]

                        = -(AB-BA)

∴(AB-BA) is a skew symmetric matrix.

∴ The option (A) is correct.

Question 12. If A =Solutions Class 12 maths Chapter-3 (matrices)and A + A′ = I, then the value of α is

(A)π/6    (B) π/3

(C) π    (D)3π/2

Solution:

Solutions Class 12 maths Chapter-3 (matrices)

Solutions Class 12 maths Chapter-3 (matrices)

On comparing both sides, we get

           2cosα = 1

⇒      cosα = \frac{1}{2}

⇒      cosα = cos\frac{π}{3}

⇒      α = \frac{π}{3}

∴ The option (B) is correct.

Chapter-3 (matrices)

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