NCERT Solutions for Class 12 Maths Chapter 3 – Matrices Ex 3.3
Page No 88:
Question 1:
Find the transpose of each of the following matrices:
(i)
(ii)
(iii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_44a1cbaa.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_m57c996fe.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6298/Chapter%203_html_m593e88d6.gif)
![](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)
![](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)
![](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)
![](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)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_ma86f68.gif)
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6299/Chapter%203_html_m38322244.gif)
![](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)
![](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)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_m75e2ee23.gif)
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_47ebe695.gif)
![](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)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_5ba3b1e5.gif)
Answer:
(i) It is known that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6303/Chapter%203_html_26ccd7a4.gif)
![](https://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)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6306/Chapter%203_html_2dfc323.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6306/Chapter%203_html_m59a7e5b7.gif)
Answer:
We know that![](https://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_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)′ =
where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6308/Chapter%203_html_4c8a1f1.gif)
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6308/Chapter%203_html_m7e3675d5.gif)
![](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)
![](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)
Page No 89:
Question 6:
If (i)
, then verify that ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_71f9fb25.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_m83faea8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_71f9fb25.gif)
(ii)
, then verify that ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_71f9fb25.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6311/Chapter%203_html_m1255699a.gif)
![](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
is a symmetric matrix
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6312/Chapter%203_html_m9e43fbb.gif)
(ii) Show that the matrix
is a skew symmetric matrix
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6312/Chapter%203_html_54726259.gif)
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 matrix
, verify that
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_37ef6a6d.gif)
(i)
is a symmetric matrix
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_543c8552.gif)
(ii)
is a skew symmetric matrix
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6315/Chapter%203_html_35ceded.gif)
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_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)
(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_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)
Question 9:
Find
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6317/Chapter%203_html_m53402729.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6317/Chapter%203_html_m5cace21a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6317/Chapter%203_html_m53356aff.gif)
Answer:
The given matrix is![](https://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_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)
![](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)
![](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)
![](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)
![](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,
is a symmetric matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m36871e07.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m4c2ce988.gif)
Thus,
is a skew-symmetric matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_734cc2e.gif)
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,
is a symmetric matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m36871e07.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m7f24fec9.gif)
Thus,
is a skew-symmetric matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_734cc2e.gif)
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,
is a symmetric matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m36871e07.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_30d8c7a6.gif)
Thus,
is a skew-symmetric matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_734cc2e.gif)
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,
is a symmetric matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m36871e07.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_m3103772f.gif)
Thus,
is a skew-symmetric matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6318/Chapter%203_html_734cc2e.gif)
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)
Page No 90:
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
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:
If
, then
, if the value of α is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_5dbc08c7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_48837b3.gif)
A.
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_m4e8d241e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/232/6324/Chapter%203_html_m217ed0f1.gif)
![](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)
![](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)