Find the best tutors and institutes for Class 12 Tuition
Q1:
If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
Solution :
Given Variables & Initial Setup
Let $A$ and $B$ be two symmetric matrices of the same order $n \times n$.
By the definition of symmetric matrices [Definition of Symmetric Matrices]:
- $A^T = A$
- $B^T = B$
We are required to prove that the matrix $X = AB - BA$ is a skew-symmetric matrix. A matrix $X$ is skew-symmetric if and only if $X^T = -X$ [Definition of Skew-Symmetric Matrices].
Step 1: Applying the Transpose Operation
To determine the nature of $X = AB - BA$, we take the transpose of the entire expression:
$X^T = (AB - BA)^T$
Using the linearity property of the transpose operation, i.e., $(A - B)^T = A^T - B^T$ [Property of Transpose]:
$X^T = (AB)^T - (BA)^T$
Step 2: Expanding via Reversal Law
Utilizing the reversal law of transpose, which states $(PQ)^T = Q^T P^T$ [Reversal Law of Matrix Transpose]:
$X^T = (B^T A^T) - (A^T B^T)$
Step 3: Substitution and Simplification
Substitute the given symmetric properties ($A^T = A$ and $B^T = B$) into the expression:
$X^T = (BA) - (AB)$
To factor out a negative sign and align with the original matrix $X = AB - BA$:
$X^T = -(AB - BA)$
$X^T = -X$
Analytical Conclusion
Since we have demonstrated that $(AB - BA)^T = -(AB - BA)$, the matrix satisfies the fundamental condition for skew-symmetry [Definition of Skew-Symmetric Matrices].
Final Solution: Since the transpose of the matrix (AB - BA) is equal to its negative, the matrix AB - BA is proved to be a skew-symmetric matrix.
More Questions from Class 12 Mathematics Matrices Miscellaneous Exercise on Chapter 3
- Q2: Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
- Q3: Find the values of $x, y, z$ if the matrix $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfy the equation $A′A = I$.
- Q4: For what values of $x$ : $\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = O$?
- Q5: If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 – 5A + 7I = 0$.
- Q6: Find $x$, if $\begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = O$
- Q7(a): A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: (a) If unit sale prices of $x, y$ and $z$ are ` 2.50, ` 1.50 and ` 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
- Q7(b): A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: (b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.
- Q8: Find the matrix $X$ so that $X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$
- Q9: If $A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ is such that $A^2 = I$, then
- Q10: If the matrix A is both symmetric and skew symmetric, then
- Q11: If A is square matrix such that $A^2 = A$, then $(I + A)^3 – 7 A$ is equal to
CBSE Solutions for Class 12 Mathematics Matrices
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Matrices
Top Tutor
Venkatraman Nagar Chennai, Chennai
10 yrs of
Exp
26 students
4290 hrs of classes completed
400
per hour
Online Classes
I am a software engineer working in MNC company. However i do like to teach maths and science subjects. I have a good experience in teaching maths and science and great in make them understanding the concepts in a most practical way and made them to score high marks.
The sessions are very interactive, he provides lots of worksheets and Test papers to strengthen the concepts. He doesn't rush through any topic and is ready to clear our doubts even when asked multiple times.
Top Tutor
A B Road, Indore
4 yrs of
Exp
Online Classes
Student's Home
Tutor's home
I am a professional tutor with four years of experience in teaching. Having a degree in M.Com from Vikram University and an MBA from IMD Indore I have also completed my B.Ed from IPS Academy Indore in the year 2015. I teach Spoken English, Accountancy, Business Studies, Economics, Statistics, Secretarial Practices, etc., for State, CBSE, ISC/ICSE Board students. I am well versed in English, Hindi and can take classes at Student's Home, My Home and Online Classes via video chat, skype, google hangout, etc.
Top Tutor
HSR Layout, Bangalore
7 yrs of
Exp
2 students
500
per hour
Online Classes
I am an experienced, qualified teacher and tutor with over 4 years of experience in teaching Maths,Physics and Chemistry, across different boards including CBSE, ICSE, IGCSE and State Board. Passionate about solving mathematical problems and chemical equations, over the years I have helped hundreds of students overcome their fear of Maths,Physics and Chemistry. So far, I have worked as a Full Time Tutor and guided over 189 students reach their goals in the respective subjects
Manohar is a thorough professional and student-friendly tutor. He has generated a lot of interest in my son on studies and has always made it very easy my son to understand the concepts and kept him very motivated to push himself beyond. My son is in 12th IB and we have engaged Me Manohar for Math, Chem and Phy HL and we would continue to engage him for college as well. We highly recommend Mr Manohar for 12 th grade and engineering basis our experience.
Top Tutor
Edapally, Kochi
4 yrs of
Exp
48 hrs of classes completed
500
per hour
Student's Home
Tutor's home
Qualified french teacher and tutor with a master degree in french from delhi university. Working as a language expert in ernst and young. Qualified. Have 4 years of teaching experience. With c1 certification.
Super Tutor
Sector 83, Gurgaon
20 yrs of
Exp
103 students
5027 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.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Matrices Miscellaneous Exercise on Chapter 3 worksheets
Download Now
