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Exercise 3.3 - Chapter 3 Matrices class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Find the transpose of each of the following matrices:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_m57c996fe.gif  (ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_m593e88d6.gif  (iii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_44a1cbaa.gif

Answer:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_335879ce.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_m29f3400.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_m446386d0.gif

Question 2:

If

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_47f1de0a.gif andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_ma86f68.gif , then verify that

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_m38322244.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_1114f29a.gif

Answer:

We have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_34037adf.gif

(i)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_57391567.gif

(ii)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_169dfbbc.gif

Question 3:

If

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_26943bbc.gif andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_m75e2ee23.gif , then verify that

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_47ebe695.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_5ba3b1e5.gif

Answer:

(i) It is known thathttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_26ccd7a4.gif

Therefore, we have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_6fc9bbf0.gif

(ii)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_7f0d2fa4.gif

Question 4:

If

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6306/Chapter%203_html_m4cf4b83b.gif andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6306/Chapter%203_html_2dfc323.gif , then find https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6306/Chapter%203_html_m59a7e5b7.gif

Answer:

We know thathttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6306/Chapter%203_html_26ccd7a4.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6306/Chapter%203_html_83a5b10.gif

Question 5:

For the matrices A and B, verify that (AB)′ = https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6308/Chapter%203_html_4c8a1f1.gif  where

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6308/Chapter%203_html_m7e3675d5.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6308/Chapter%203_html_m4e043813.gif

Answer:

(i)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6308/Chapter%203_html_3a781833.gif

(ii)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6308/Chapter%203_html_m784b9bad.gif

Question 6:

If (i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_m83faea8.gif , then verify that https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_71f9fb25.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_m1255699a.gif , then verify that https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_71f9fb25.gif

Answer:

(i)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_m6e30d5bf.gif

(ii)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_m3be39a84.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_59300ef7.gif

Question 7:

(i) Show that the matrix https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6312/Chapter%203_html_m9e43fbb.gif is a symmetric matrix

(ii) Show that the matrix https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6312/Chapter%203_html_54726259.gif is a skew symmetric matrix

Answer:

(i) We have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6312/Chapter%203_html_m1551fd15.gif

Hence, A is a symmetric matrix.

(ii) We have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6312/Chapter%203_html_m1af745ec.gif

Hence, A is a skew-symmetric matrix.

Question 8:

For the matrixhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_37ef6a6d.gif , verify that

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_543c8552.gif  is a symmetric matrix

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_35ceded.gif  is a skew symmetric matrix

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_m7e4b395d.gif

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_m10ec19fa.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_m682e2c84.gif

Hence,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_543c8552.gif  is a symmetric matrix.

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_69093134.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_5c7ba90.gif

Hence,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_35ceded.gif is a skew-symmetric matrix.

Question 9:

Find

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6317/Chapter%203_html_m53402729.gif andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6317/Chapter%203_html_m5cace21a.gif , when https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6317/Chapter%203_html_m53356aff.gif

Answer:

The given matrix ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6317/Chapter%203_html_m53356aff.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6317/Chapter%203_html_27ca0a22.gif

Question 10:

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

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m342819bf.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_2ec2379e.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_269d865f.gif

(iv) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m199cdd3.gif

Answer:

(i)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m3e2a51ac.gif

Thus, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m36871e07.gif  is a symmetric matrix.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m4c2ce988.gif

Thus, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_734cc2e.gif  is a skew-symmetric matrix.

Representing A as the sum of P and Q:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m1651ee6.gif

(ii)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_362b1652.gif

Thus, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m36871e07.gif  is a symmetric matrix.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m7f24fec9.gif

Thus, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_734cc2e.gif  is a skew-symmetric matrix.

Representing A as the sum of P and Q:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m40160919.gif

(iii)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m3dd9318b.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_66166e02.gif

Thus, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m36871e07.gif  is a symmetric matrix.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_30d8c7a6.gif

Thus, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_734cc2e.gif  is a skew-symmetric matrix.

Representing A as the sum of P and Q:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m171b504f.gif

(iv)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_ff2e57e.gif

Thus, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m36871e07.gif  is a symmetric matrix.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m3103772f.gif

Thus, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_734cc2e.gif is a skew-symmetric matrix.

Representing A as the sum of P and Q:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_55f000b6.gif

Question 11:

If AB are symmetric matrices of same order, then AB − BA is a

A. Skew symmetric matrix B. Symmetric matrix

C. Zero matrix D. Identity matrix

Answer:

The correct Answer is A.

A and B are symmetric matrices, therefore, we have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6320/Chapter%203_html_10b057d5.gif

Thus, (AB − BA) is a skew-symmetric matrix.

Question 12:

Ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_5dbc08c7.gif , thenhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_48837b3.gif , if the value of α is

A. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_m217ed0f1.gif  B. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_m4e8d241e.gif

C. π D. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_1549eb8f.gif

Answer:

The correct Answer is B.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_ma75ac42.gif

Comparing the corresponding elements of the two matrices, we have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_m45490ea.gif

Also Read : Exercise-3.4-Chapter-3-Matrices-class-12-ncert-solutions-Maths

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