Question

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 1

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.