Toppers Study Toppers Study Toppers StudyDetailed NCERT Solutions for 12 Mathematics I 3. Matrices to simplify learning. Understand chapters clearly and practice with free solutions for better results.
Detailed NCERT Solutions for 12 Mathematics I 3. Matrices to simplify learning. Understand chapters clearly and practice with free solutions for better results.
Preparing for exams becomes easier with Exercise 3.3. Whether you are studying for board exams or mid-term exams, 12 Mathematics I Chapter 3. Matrices solutions provide quick revising points, well-structured answers, and additional practice material to help you score better.
ncert_solutionsExercise 3.3
Ques.1. Find the transpose of each of the following matrices:
(i) 
(ii) 
(iii) 
Ans. (i) Let A =
Transpose of A = A’ or AT = 
(ii)
Transpose of A = A’ or AT = 
(iii)
Transpose of A = A’ or AT = 
Ques.2. If A = 
and B = 
then verify that:
(i) 
(ii) 
Ans. (i) A + B = 
= 
= 
L.H.S. = (A + B)’ = 
= 
R.H.S. = A’ + B’ = 
= 
= 
=
L.H.S. = R.H.S.
hance Proved.
(ii) A – B = 
= 
= 
L.H.S. = (A – B)’ = 
= 
R.H.S. = A’ – B’ = 
= 
= 
=
L.H.S. = R.H.S.
hance Proved.
and B = 
then verify that:(i)
(ii)
Ans. Given: A’ = and B = then (A’)’ = A = 
(i) A + B = 
= 
L.H.S. = (A + B)’ = 
R.H.S. = A’ + B’
= 
= 
= 
= 
L.H.S. = R.H.S.
hance Proved.
(ii) A – B = 
= 
L.H.S. = (A – B)’
= 
R.H.S. = A’ – B’
= 
= 
= 
= 
L.H.S. = R.H.S.
hance Proved.
and B = 
then find (A + 2B)’.Ans. Given: A’ = and B = then (A’)’ = A = 
A +2B = 
= 
]
= 
= 
(A + 2B)’ = 
(i) A = 
B = 
(ii) A = 
B = 
Ans. (i) AB = 
= 
L.H.S. = (AB)’
= 
= 
R.H.S. = B’A’
= 
= 
=
L.H.S. = R.H.S.
hance Proved.
(ii) AB = 
= 
L.H.S. = (AB)’
= 
= 
R.H.S. = B’A’
= 
= 
=
L.H.S. = R.H.S.
hance Proved.
then verify that A’A = I.(ii) If A = 
then verify that A’A = I.
Ans. (i) L.H.S. = A’A = 
= 
= 
= 
= I = R.H.S.
(ii) L.H.S. = A’A
= 
= 
= 
=
= I = R.H.S.
is a symmetric matrix.(ii) Show that the matrix A = 
is a skew symmetric matrix.
Ans. (i) Given: A = ……….(i)
Changing rows of matrix A as the columns of new matrix A’ = = A
A’ = A
hance, by definitions of symmetric matrix, A is a symmetric matrix.
(ii) Given: A = ……….(i)
A’ = 
= 
Taking 
common, A’ = 
= – A [From eq. (i)]
hance, by definition matrix A is a skew-symmetric matrix
verify that:(i) (A + A’) is a symmetric matrix.
(ii) (A – A’) is a skew symmetric matrix.
Ans. (i) Given: A = 
Let B = A + A’ = 
= 
= 
B’ = 
= B
B = A + A’ is a symmetric matrix.
(ii) Given:
Let B = A – A’ = 
= 
= 
B’ = 
Taking common, 
= – B
B = A – A’ is a skew-symmetric matrix.
(A + A’) and (A – A’) when A = 
Ans. Given: A = 
A’ = 
Now, A + A’ = 
= 
= 
(A + A’) = 
Now, A – A’ = 
= 
= 
(A – A’) = 
= 
(i) 
(ii) 
(iii) 
(iv) 
Ans. (i) Given: A = so, A’ = 
Symmetric matrix = (A + A’)
= 
= 
= 
Skew symmetric matrix = (A – A’)
= 
= 
= 
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(ii) Given: A = so, A’ =
Symmetric matrix = (A + A’)
= 
= 
=
And Skew symmetric matrix = (A – A’)
= 
= 
=
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(iii) Given: A = so, A’ = 
Symmetric matrix = (A + A’)
= 
= 
= 
And Skew symmetric matrix = (A–A’)
= 
= 
= 
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(iv) Given: A = so, A’ = 
Symmetric matrix = (A + A’)
= 
= 
= 
And Skew symmetric matrix = (A – A’)
= 
= 
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
Ques.11. If A and B are symmetric matrices of same order, AB – BA is a:
(A) Skew-symmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(S) Identity matrix
Ans. Given: A and B are symmetric matrices 
A = A’ and B = B’
Now, (AB – BA)’ = (AB)’ – (BA)’
(AB – BA)’ = B’A’ – A’B’ [Reversal law]
(AB – BA)’ = BA – AB [From eq. (i)]
(AB – BA)’ = – (AB – BA)
(AB – BA) is a skew matrix.
hance, option (A) is correct.
, then A + A’ = I, if the value of 
is:(A) 
(B) 
(C) 
(D) 
Ans. Given: A = 
Also A + A’ = I
Equating corresponding entries, we have
hance, option (B) is correct.
Exercise 3.3 are created by experts to give step-by-step explanations. Around 60–70% of exam questions are based on NCERT concepts. Our 12 Mathematics I Chapter 3. Matrices solutions help you understand the core concepts and practice effectively.
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Our Exercise 3.3 are useful for both board exams and mid-term exams. For 12 Mathematics I, we provide notes, exercises, and important Q&A so that you can revise smartly and write perfect answers in exams.
In short, Exercise 3.3 for 12 Mathematics I Chapter 3. Matrices are a complete study package. With quick revising points, NCERT exercises, and additional important questions, you can prepare effectively for exams. Make these solutions your study companion and excel in your academic journey.
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