
If \[A\] is a square matrix, then which of the following matrices is not symmetric
A. \[A + {A^\prime }\]
B. \[A{A^\prime }\]
C. \[{A^\prime }A\]
D. \[A - {A^\prime }\]
Answer
161.1k+ views
Hint: For a matrix to be symmetric then a square matrix must be equal to its transpose matrix. As a result, a symmetric and skew symmetric matrix is both square matrices. The distinction between them is that the symmetric matrix is equal to its transpose, whereas the skew symmetric matrix is equal to its negative. Since equal matrices have equal dimensions, only square matrices can be symmetric. A symmetric matrix's elements are symmetric with regard to the main diagonal.
Formula Used:
If matrix A is symmetric, then
\[A = {A^T}\]
Complete Step-by-Step solution: We have been provided in the question that,
\[A\] is a square matrix
And we are to find from the given which is not symmetric.
If considered that \[{\rm{A}}\]is a square matrix, then \[{{\rm{A}}^\prime }\]represents its transpose matrix
Then we understood that \[{\rm{A}} + {{\rm{A}}^\prime }\] is symmetric and \[{\rm{A}} - {{\rm{A}}^\prime }\] is skewing symmetric.
Now, we can write the matrix A as
\[A = \left( {\dfrac{{A + {A^\prime }}}{2}} \right) + \left( {\dfrac{{A - {A^\prime }}}{2}} \right)\]
Therefore, of all the above matrix,
We came to a conclusion that \[{\rm{A}} - {{\rm{A}}^\prime }\]is not symmetric.
Therefore, if \[A\] is a square matrix then \[A - {A^\prime }\] is not symmetric.
Hence, the option D is correct
Note: Student should remember that if and only if A = A', ‘A’ is symmetric. The only fundamental difference between a symmetric and skew symmetric matrix is that the symmetric matrix's transpose equals the original matrix. The skew symmetric matrix's transpose equals the negative of the original matrix.
Formula Used:
If matrix A is symmetric, then
\[A = {A^T}\]
Complete Step-by-Step solution: We have been provided in the question that,
\[A\] is a square matrix
And we are to find from the given which is not symmetric.
If considered that \[{\rm{A}}\]is a square matrix, then \[{{\rm{A}}^\prime }\]represents its transpose matrix
Then we understood that \[{\rm{A}} + {{\rm{A}}^\prime }\] is symmetric and \[{\rm{A}} - {{\rm{A}}^\prime }\] is skewing symmetric.
Now, we can write the matrix A as
\[A = \left( {\dfrac{{A + {A^\prime }}}{2}} \right) + \left( {\dfrac{{A - {A^\prime }}}{2}} \right)\]
Therefore, of all the above matrix,
We came to a conclusion that \[{\rm{A}} - {{\rm{A}}^\prime }\]is not symmetric.
Therefore, if \[A\] is a square matrix then \[A - {A^\prime }\] is not symmetric.
Hence, the option D is correct
Note: Student should remember that if and only if A = A', ‘A’ is symmetric. The only fundamental difference between a symmetric and skew symmetric matrix is that the symmetric matrix's transpose equals the original matrix. The skew symmetric matrix's transpose equals the negative of the original matrix.
Recently Updated Pages
If tan 1y tan 1x + tan 1left frac2x1 x2 right where x frac1sqrt 3 Then the value of y is

Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2026 Syllabus PDF - Download Paper 1 and 2 Syllabus by NTA

JEE Main Eligibility Criteria 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

JEE Advanced 2025 Notes
