Find the best tutors and institutes for Class 12 Tuition
Q10:
If the matrix A is both symmetric and skew symmetric, then
Solution :
Step 1: Establishing Definitions
To determine the nature of a matrix $A$ that is simultaneously symmetric and skew-symmetric, we must apply the foundational definitions of these matrix classifications [Per the definitions of square matrices]:
- Symmetric Matrix: A square matrix $A$ is symmetric if $A^T = A$.
- Skew-Symmetric Matrix: A square matrix $A$ is skew-symmetric if $A^T = -A$.
Step 2: Equating the Conditions
Given that the matrix $A$ satisfies both conditions simultaneously, we set the right-hand sides of the two definitions equal to each other:
$A = A^T$ (from the property of symmetric matrices)
$A^T = -A$ (from the property of skew-symmetric matrices)
By the Transitive Property of Equality [By Mathematical Axioms], we substitute $A^T$ in the first equation with $-A$:
$A = -A$
Step 3: Algebraic Manipulation
To solve for $A$, we perform the following algebraic steps:
$A + A = 0$ [Adding $A$ to both sides of the equation]
$2A = O$ [Where $O$ denotes the zero matrix of the same order as $A$]
$A = \frac{1}{2}O$
$A = O$
Step 4: Logical Deduction of the Matrix Elements
If $A = O$, then every element $a_{ij}$ within the matrix $A$ must be zero for all $i, j$. A square matrix where all entries are zero is defined as a Null Matrix or Zero Matrix.
Step 5: Theoretical Conclusion
Any matrix $A$ that is both symmetric and skew-symmetric must necessarily contain only zero elements. If even one element were non-zero (e.g., $a_{ij} = k \neq 0$), then $a_{ij} = k$ would imply $a_{ji} = k$ (symmetric) and $a_{ji} = -k$ (skew-symmetric), forcing $k = -k$, which is only possible if $k=0$ [Per the properties of field characteristics where $char \neq 2$].
Final Solution: A is a zero matrix.
More Questions from Class 12 Mathematics Matrices Miscellaneous Exercise on Chapter 3
- Q1: If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
- 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
- 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
Super Tutor
Banjara Hills, Hyderabad
12 yrs of
Exp
37 students
1639 hrs of classes completed
800
per hour
Online Classes
Tutor's home
Although I haven't taught in schools or colleges, I have been invited to deliver guest lectures at reputed universities. These sessions allowed me to share my expertise with advanced learners and contribute to academic discussions. My focus has always been on delivering value through practical insights, making complex topics accessible and engaging for university students.
The experience is very good , give an detailed explanation with real life examples and also makes the concept easy to understand.
Super Tutor
Vesu, Surat
20 yrs of
Exp
14 students
2475 hrs of classes completed
500
per hour
Online Classes
I have experience teaching Mathematics to Class 12 students, focusing on building strong conceptual understanding and problem-solving skills. I teach key topics such as Calculus (limits, derivatives, and integrals), Matrices, Determinants, Probability, and Vectors. My teaching approach includes explaining concepts step-by-step, solving examples on the board, and giving practice questions similar to board exam patterns. I regularly conduct revision tests, doubt-clearing sessions, and discuss previous years’ question papers to help students prepare effectively for board examinations. I encourage students to understand the logic behind formulas rather than memorizing them. My goal is to make mathematics easier, improve students’ confidence, and help them achieve good results in Class 12 board exams.
A good teacher who relates to the child n guides them to perform better. Has clarity n mastery over the subject he teaches.
Super Tutor
Kasavanahalli Main Rd Kasavanahalli, Bangalore
2 yrs of
Exp
6 students
897 hrs of classes completed
800
per hour
Online Classes
I am a teacher. I love teaching and want to share my knowledge with my students. I will take online classes. My approach combines clear concepts, regular practice, and student-friendly strategies to make learning effective and enjoyable.
"Found it to be a very intuitive learning methodology, practical approach, helpful in understanding."
Super Tutor
Panvel, Mumbai
12 yrs of
Exp
21 students
3291 hrs of classes completed
500
per hour
Online Classes
I provide private tuition for Physics and Mathematics for Class XI and XII (All Boards) Physics for Engineering Aspirants Physics for Medical Aspirants Class IX and X Mathematics and Science.
I am grateful that my son have been taught by you. Your dedication, knowledge, and passion for teaching make every class engaging and productive. You explain complex concepts in a simple and understandable manner. Thank you Sir .
Top Tutor
Rajarajeshwari Nagar, Bangalore
12 yrs of
Exp
51 students
1960 hrs of classes completed
500
per hour
Online Classes
Student's Home
I have been training students for 12th Board examinations for more than a decade now.
He is outstanding as a teacher, knowledgeable and dedicated. He has helped my daughter so much with his knowledge, that the girl who was just running away from accounts is getting more than 90 percent
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Matrices Miscellaneous Exercise on Chapter 3 worksheets
Download Now
