Question

If $A$ is any square matrix, then $\left( A+{{A}^{T}} \right)$ is a ............... matrix. Fill in the blank:
A. symmetric
B. skew symmetric
C. scalar
D. identity

Answer 1

Hint: For this question we will first see what a square matrix is then we will assume a square matrix of any order (preferably 3) then we will write its transpose by changing rows of the original into the columns of the new matrix and vice-a-versa. Finally we will put both the original and transposed matrix into the given condition and find out the type of the resultant matrix.

Complete step by step answer:
First, we know that a square matrix is a matrix with the same number of rows and columns. That means a $n\times n$ matrix is known as a square matrix of order$n$.
Given that, $A$ is a square matrix.
Now, let matrix $A=\left[ \begin{matrix}
   a & b & c \\
   d & f & g \\
   e & h & i \\
\end{matrix} \right]$
Now ${{A}^{T}}$ , stands for transpose of $A$ . Transpose of a matrix means a new matrix whose rows are the columns of the original matrix which means the columns of the new matrix are the rows of the original.
Therefore:
 $\begin{align}
  & {{A}^{T}}={{\left[ \begin{matrix}
   a & b & c \\
   d & f & g \\
   e & h & i \\
\end{matrix} \right]}^{T}}=\left[ \begin{matrix}
   a & d & e \\
   b & f & h \\
   c & g & i \\
\end{matrix} \right] \\
 & {{A}^{T}}=\left[ \begin{matrix}
   a & d & e \\
   b & f & h \\
   c & g & i \\
\end{matrix} \right] \\
\end{align}$

Now, it is given in the question that we have to find out what type of matrix $\left( A+{{A}^{T}} \right)$ is?
Now, putting the values of $A$ and ${{A}^{T}}$ in $\left( A+{{A}^{T}} \right)$:
$\begin{align}
  & A+{{A}^{T}}=\left[ \begin{matrix}
   a & b & c \\
   d & f & g \\
   e & h & i \\
\end{matrix} \right]+\left[ \begin{matrix}
   a & d & e \\
   b & f & h \\
   c & g & i \\
\end{matrix} \right]=\left[ \begin{matrix}
   2a & \left( b+d \right) & \left( c+e \right) \\
   \left( d+b \right) & 2f & \left( g+h \right) \\
   \left( e+c \right) & \left( h+g \right) & 2i \\
\end{matrix} \right] \\
 & A+{{A}^{T}}=\left[ \begin{matrix}
   2a & \left( b+d \right) & \left( c+e \right) \\
   \left( b+d \right) & 2f & \left( g+h \right) \\
   \left( c+e \right) & \left( g+h \right) & 2i \\
\end{matrix} \right] \\
\end{align}$
We can see the symmetry of $\left( A+{{A}^{T}} \right)$ about its diagonal. Therefore it is a symmetric matrix.

So, the correct answer is “Option A”.

Note: Note that the sum and difference of two symmetric matrices is again symmetric.The resultant matrix is also called as persymmetric matrix which is a square matrix which is symmetric in the northeast-to-southwest diagonal. If the diagonal elements were all zeroes then it would have qualified as a skew symmetric matrix, in skew symmetric matrix: $A=-{{A}^{T}}$ .