Find the best tutors and institutes for Class 12 Tuition
Q11:
If $A, B$ are symmetric matrices of same order, then $AB – BA$ is a
Solution :
Step 1: Establishing Definitions and Properties
Given that $A$ and $B$ are symmetric matrices of the same order $n \times n$, by definition, their transposes are equal to the matrices themselves:
- $A^T = A$ [Definition of Symmetric Matrix]
- $B^T = B$ [Definition of Symmetric Matrix]
We are required to determine the nature of the matrix $C$, defined as:
$C = AB - BA$
Step 2: Testing for Symmetry or Skew-Symmetry
To classify the matrix $C$, we evaluate its transpose $C^T$ using the properties of matrix transpose operations:
$C^T = (AB - BA)^T$
Applying the linearity property of transposes, $(X - Y)^T = X^T - Y^T$ [Property of Transpose]:
$C^T = (AB)^T - (BA)^T$
Applying the reversal law of transposes, $(AB)^T = B^T A^T$ [Property of Transpose]:
$C^T = B^T A^T - A^T B^T$
Step 3: Algebraic Substitution and Simplification
Substitute the symmetry conditions ($A^T = A$ and $B^T = B$) into the expression:
$C^T = (B)(A) - (A)(B)$
$C^T = BA - AB$
Factor out the negative sign to relate it back to the original matrix $C$:
$C^T = -(AB - BA)$
$C^T = -C$
Step 4: Theoretical Conclusion
By definition, a matrix $M$ is skew-symmetric if $M^T = -M$ [Definition of Skew-Symmetric Matrix]. Since we have demonstrated that $(AB - BA)^T = -(AB - BA)$, the matrix $AB - BA$ satisfies the criteria for a skew-symmetric matrix.
Final Solution: The matrix $AB – BA$ is a skew-symmetric matrix.
More Questions from Class 12 Mathematics Matrices EXERCISE 3.3
- Q1(i): Find the transpose of each of the following matrices: (i) $\begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}$
- Q1(ii): Find the transpose of each of the following matrices: (ii) $\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$
- Q1(iii): Find the transpose of each of the following matrices: (iii) $\begin{bmatrix} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{bmatrix}$
- Q2(i): If $A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}$, then verify that (i) $(A + B)' = A' + B'$
- Q2(ii): If $A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}$, then verify that (ii) $(A – B)' = A' – B'$
- Q3(i): If $A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, then verify that (i) $(A + B)' = A' + B'$
- Q3(ii): If $A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, then verify that (ii) $(A – B)' = A' – B'$
- Q4: If $A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$, then find $(A + 2B)'$
- Q5(i): For the matrices A and B, verify that $(AB)' = B'A'$, where (i) $A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix}$, $B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}$
- Q5(ii): For the matrices A and B, verify that $(AB)' = B'A'$, where (ii) $A = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 5 & 7 \end{bmatrix}$
- Q6(i): If (i) $A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}$, then verify that $A' A = I$
- Q6(ii): If (ii) $A = \begin{bmatrix} \sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha \end{bmatrix}$, then verify that $A' A = I$
- Q7(i): (i) Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.
- Q7(ii): (ii) Show that the matrix $A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}$ is a skew symmetric matrix.
- Q8(i): For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, verify that (i) $(A + A')$ is a symmetric matrix
- Q8(ii): For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, verify that (ii) $(A – A')$ is a skew symmetric matrix
- Q9: Find $\frac{1}{2}(A + A')$ and $\frac{1}{2}(A - A')$, when $A = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}$
- Q10(i): Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) $\begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$
- Q10(ii): Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (ii) $\begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$
- Q10(iii): Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (iii) $\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}$
- Q10(iv): Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (iv) $\begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}$
- Q12: If $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$, and $A + A' = I$, then the value of $\alpha$ is
CBSE Solutions for Class 12 Mathematics Matrices
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Matrices
Top Tutor
Taj Bagh, Nagpur
10 yrs of
Exp
2 students
119 hrs of classes completed
600
per hour
Online Classes
Student's Home
I have twelve years of experience in teaching mathematics. I hold a B.Tech in Mechanical Engineering and an M.Tech in Mechanical Engineering Design. One of my most rewarding experiences as a teacher was when I helped a student who was depressed due to poor math grades. The student's parents approached me for help, and after taking classes from me, she cleared her tenth board exam with excellent marks in math. My main passion is to teach the basics of the subject, as I've noticed that many students need more foundational knowledge and help with math. I teach Mathematics for classes eighth to twelfth through UrbanPro, both online and offline, in individual and group settings. However, my primary focus is on teaching twelfth-class students. I take great care in designing my lesson plans, ensuring students have a solid understanding of the fundamentals before moving on to more complex topics. During my classes, I use a whiteboard to visually explain concepts and engage students in an interactive learning experience. My teaching philosophy revolves around providing a strong foundation in mathematics. A solid understanding of the basics is crucial for students to excel in the subject. I am patient and understanding towards students who may struggle with math, and I employ various teaching techniques to cater to different learning styles. I need to create a positive and supportive learning environment where students feel comfortable asking questions and seeking clarification. My expertise in mechanical engineering adds dimension to my teaching, as I can provide real-world applications of mathematical concepts, making the subject more relevant and exciting for students. I am passionate about inspiring a love for mathematics in my students and helping them build confidence in their math skills. My focus on teaching the basics, creating a positive learning environment, and providing real-world applications of mathematical concepts, makes me an effective and successful teacher.
He is a talented person who can teach both maths and physics. He always teach the concept from indepth and gives boards writing format. Also as its a one to one tutoring your gonna get max attention as possible and all your doubts are going to be cleared by him at the spot. Hope it helps !!
Super Tutor
Sector 83, Gurgaon
20 yrs of
Exp
103 students
5028 hrs of classes completed
800
per hour
Online Classes
Tutor's home
With 26 years of extensive experience in teaching mathematics, I specialize in instructing students from grades 9 through 12 across various boards, including CBSE, ICSE, and State Boards. My academic qualifications include a BSc, BTech in Mathematics, MSc, BEd, and I am currently pursuing a PhD. This strong educational background, coupled with my long-standing commitment to teaching, has allowed me to develop a comprehensive approach to mathematics education. I have held prestigious positions as a College Associate Professor at esteemed institutions such as Ganga Institute of Technology and Management, VAISH College, and BLS University. My dedication to teaching has been recognized with the Best Teacher Award from both VAISH College and Ganga Institute of Technology and Management. These accolades reflect my commitment to delivering high-quality education and fostering academic excellence in my students. Currently, I run my own offline center where I conduct mathematics classes for students in grades 9 through 12. In addition to my offline classes, I also offer online sessions to provide flexible learning options. My teaching methods involve practical sessions using a whiteboard to facilitate a clear and engaging learning experience. I emphasize the importance of regular assessments, providing my students with notes, assignments, and weekly tests to track their progress and ensure thorough understanding. My approach to teaching mathematics is rooted in practical application and interactive learning. By focusing on real-world problems and fostering a supportive learning environment, I aim to help students build a strong foundation in mathematics, achieve academic success, and develop a lifelong appreciation for the subject.
Very nice teacher I saw significant improvement in my mathematics. Very carefully planned lessons. Plans realy do work. Alot of practice which helps improve overall.
Top Tutor
HMT Estate 2nd Stage, Bangalore
20 yrs of
Exp
37 students
2766 hrs of classes completed
600
per hour
Online Classes
I am a teacher. Teaching chemistry for the past 20+ years. I also have some experience in research in applied chemistry. I have a masters degree in chemistry and a bachelors degree in education. In addition to teaching I have experience in school administration and content development. My key skills are to train the students for various competitive exams like JEE, NEET, BITSAT, CET.
She is very good at clearing concepts. My son went to her for the past 2 years. Patient and articulate, I recommend her as a teacher to all those who want to get a good foundation in class XI and class XII chemistry.
Top Tutor
Virugambakkam, Chennai
20 yrs of
Exp
46 students
1713 hrs of classes completed
600
per hour
Online Classes
Hi! I’m Vinotha Arulkumar, a passionate Maths Teacher from Chennai. My qualifications are M. Sc., M. Phil. and B. Ed in Mathematics. From my 10th standard onwards, I got many centum in maths. I’m the topper of my batch in my Under Graduation, M. Phil., and B. Ed. And also I’m the gold medalist in my M. Sc., Mathematics. I possess about 16 years of teaching experience in the field of education. One year in Arts and science college, seven years in Engineering colleges and taking tuition for 10th and 12th maths for more than 10 years. In Arts and Science College I handled B. Sc., Mathematics and M. Com. In Engineering colleges, I handled maths papers for BE, B. Tech., ME, M. Tech., MCA and MBA. While working in engineering colleges, I have received Proficiency Awards for 6 times as I produced 100% results in engineering mathematics. Due to the pandemic situation, presently I’m taking online classes only. I can teach in English and Tamil. My usual class duration is 1 Hr. The key element of my teaching method is to keep on interacting with students while teaching. Before getting in to a problem, usually I will make the students understand the concepts and fundamentals of the chapter first. Then, I will let the students to understand the problem and let them solve it with my guidance. So that they will never forget the concepts. I often conduct online MCQ assessments to evaluate the progress of my students. Bases on the assessment results, I will give individual attention to each student. I keep producing excellent results every year through my vast knowledge & experience in Mathematics. Lot of my students got centum marks in their board exams! Thank you and all the best for your bright future!
Good teaching skills. Explain the concepts clearly, student was able to understand easily. Good communication.
Top Tutor
HRBR Layout, Bangalore
20 yrs of
Exp
5 students
1000
per hour
Online Classes
Student's Home
Tutor's home
I have 30 years of experience in teaching IELTS and English to students in Class I-V and Class X-XII. I have worked in international schools with IB and Cambridge syllabus. I am working as Head of the ESL ( English as Second Language) department.
Wonderful, excellent teacher. Clears the concept. Well versed with IB syllabus. My son really improved his grades under her teaching.
