If a is A symmetric matrix and

Question:

If $\mathrm{a}$ is $\mathrm{A}$ symmetric matrix and $\mathrm{B}$ is a skewsymmetrix matrix such that $\mathrm{A}+\mathrm{B}=\left[\begin{array}{cc}2 & 3 \\ 5 & -1\end{array}\right]$, then $\mathrm{AB}$ is equal to :

  1. $\left[\begin{array}{cc}-4 & 2 \\ 1 & 4\end{array}\right]$

  2. $\left[\begin{array}{cc}-4 & -2 \\ -1 & 4\end{array}\right]$

  3. $\left[\begin{array}{cc}4 & -2 \\ -1 & -4\end{array}\right]$

  4. $\left[\begin{array}{ll}4 & -2 \\ 1 & -4\end{array}\right]$


Correct Option: , 3

Solution:

$\mathrm{A}=\mathrm{A}^{\prime}, \mathrm{B}=-\mathrm{B}^{\prime}$

$\mathrm{A}+\mathrm{B}=\left[\begin{array}{cc}2 & 3 \\ 5 & -1\end{array}\right]$ ...........(1)

$\mathrm{A}^{\prime}+\mathrm{B}^{\prime}=\left[\begin{array}{cc}2 & 5 \\ 3 & -1\end{array}\right]$

$A-B=\left[\begin{array}{cc}2 & 5 \\ 3 & -1\end{array}\right]$ .............(2)

After adding Eq. (1) & (2)

$A=\left[\begin{array}{cc}2 & 4 \\ 4 & -1\end{array}\right], \quad B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$

$\mathrm{AB}=\left[\begin{array}{cc}4 & -2 \\ -1 & -4\end{array}\right]$

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