Question

If a matrix A is both symmetric and skew – symmetric, then A is:
A.${\text{I}}$
B.O
C.Triangular matrix
D.Zero matrix

Answer 1

Hint: We will use the definitions of symmetric matrix and skew – symmetric matrix to solve this question for the correct option. The condition for a matrix to be a symmetric matrix is A =${A^T}$and for skew – symmetric matrix, ${A^T} = - A$.

Complete step-by-step answer:
We are given that a matrix A is both symmetric and skew – symmetric. We need to find the type of matrix A.
We will first define symmetric and skew – symmetric matrices.
Definition of symmetric matrix: In mathematics, a symmetric matrix, say A, is a square matrix which is equal to its transpose, ${A^T}$.
Or in terms of an equation, we can write A =${A^T}$.
Definition of skew – symmetric matrix: In mathematics, a skew – symmetric matrix, say A, is a square matrix whose transpose of the matrix,${A^T}$ is equal to the negative of the matrix, $ - A$.
Or in terms of an equation, we can write${A^T} = - A$.
Also, the main diagonal elements of a skew – symmetric matrix are all zero.
According to the question, A is both symmetric and skew – symmetric, therefore, we can write
$ \Rightarrow A = {A^T} = - A$ and main diagonal elements of A are zero.
Now, let us look at the options. We are required to check if the options satisfy $A = {A^T} = - A$ or not.
Option (A): $I$ = Identity matrix
For an identity matrix, $A = {A^T} = - A$can’t hold true since $I \ne - I$ .
Therefore, option (A) is not correct.
Option (B): O
It can’t be true since A is a matrix given $A = {A^T} = - A$and there is no condition specified for A to be O. therefore, option (B) is incorrect.
Option (C): Triangular matrix
In a triangular matrix, the main diagonal elements can never be zero but here, the main diagonal elements of A are zero being a skew – symmetric matrix. Hence, option (C) is also incorrect.
Option (D): Zero matrix
In a zero matrix, all the elements of the matrix are zero and this is the only matrix where $A = {A^T} = - A$can hold true since there is no negative of zero (or we can say that negative of zero is zero).
Therefore, option (D) is correct.
Note: In this question, we have only used the definition of the concepts given in the question i.e. symmetric and skew – symmetric matrix. It is necessary for you to understand the concepts since it is not a tough question to lose marks. You can also solve this question directly by observation of the given options only without explaining all the options but such questions can have multiple answers, hence, it is advisable to check every option.