Question-94

Consider these two statements:

• S1: The sum of two symmetric matrices is a symmetric matrix.
• S2: The product of two symmetric matrices is a symmetric matrix.

Select the correct option.

• S1 is true, S2 is false
• S1 is false, S2 is true
• S1 and S2 are true
• S1 and S2 are false

Answer

• S1 is true, S2 is false
• S1 is false, S2 is true
• S1 and S2 are true
• S1 and S2 are false

Solution

S1 is true and S2 is false. If \(\displaystyle A,B\) are symmetric, then:

\[ \displaystyle ( A+B)^{T} =A^{T} +B^{T} =A+B \]

A counter-example for S2:

\[ \begin{equation*} A=\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} ,\ B=\begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix} \end{equation*} \]

Then:

\[ \begin{equation*} AB=\begin{bmatrix} 1 & 1\\ 0 & 0 \end{bmatrix} \end{equation*} \]

Clearly, \(\displaystyle AB\) is not symmetric.