Question

If A is a square matrix, then
A) $A + {A^T}$ is symmetric
B) $A{A^T}$ is skew symmetric
C) ${A^T} + A$ is skew symmetric
D) ${A^T}A$ is skew symmetric

Answer 1

Hint: According to the question given in the question we have to determine the correct option for A which is a square matrix. So, first of all we have to understand about a transpose matrix which is explained as below:
Transpose matrix: The transpose of a matrix is simply a flipped version of the original matrix and we can find a transpose of a matrix by switching its rows with it columns and we can denote a transpose of a matrix A by ${A^T}$
Now, we have to take a square matrix A and then we have to determine the transpose of the matrix which can be done by opening the brackets.
Now, to solve the transpose of the matrix we have to use the identity for the matrix as mentioned below:
$ \Rightarrow {({A^T})^T} = A$…………………(1)
Which means, if we find the transpose of a transposed matrix then it will become a matrix or we can say in our case a square matrix.

Complete step-by-step solution:
Step 1: First of all we have to let a square matrix as A now, we have to find the transpose of the matrix which is as below:
$ = {A^T}$
Step 2: Now, we have to check the option (A) which is $A + {A^T}$ and now, we have to determine its transpose. Hence,
$ = {(A + {A^T})^T}$
Step 3: Now, to solve the matrix as obtained in the solution step 2 we have to open all the brackets and we have to simplify it. Hence,
$ = {A^T} + {({A^T})^T}$
Step 4: Now, to solve the matrix as obtained in the solution step 3 we have to use the identity (1) as mentioned in the solution hint.
$ = {A^T} + A$
Step 5: Now, as we have obtained the matrix which is similar to the matrix we solved.
Hence, we have obtained that if A is a square matrix then $A + {A^T}$is symmetric.

Therefore option (A) is correct.

Note: The transpose of a matrix is an operator which flips a matrix over its diagonal that is, it switches the row and column indices of the matrix A by producing another matrix, and often denoted by ${A^T}$.