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If a matrix $\mathrm{B}$ is obtained from a square matrix $\mathrm{A}$ by interchanging any two of its rows, then what is $|\mathrm{A}+\mathrm{B}|$ equal to
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Let $A=\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$
Let rows 1 and 2 be interchanged.
and $B=\left[\begin{array}{lll}\mathrm{d} & \mathrm{e} & \mathrm{f} \\ \mathrm{a} & \mathrm{b} & \mathrm{c} \\ \mathrm{g} & \mathrm{h} & \mathrm{i}\end{array}\right]$
$A+B=\left[\begin{array}{ccc}a+d & b+e & c+f \\ a+d & b+e & c+f \\ 2 g & 2 h & 2 i\end{array}\right]$
$|A+B|=\left|\begin{array}{ccc}a+d & b+e & c+f \\ a+d & b+e & c+f \\ 2 g & 2 h & 2 i\end{array}\right|$
$=0 \quad$ (since two rows are identical)
Let rows 1 and 2 be interchanged.
and $B=\left[\begin{array}{lll}\mathrm{d} & \mathrm{e} & \mathrm{f} \\ \mathrm{a} & \mathrm{b} & \mathrm{c} \\ \mathrm{g} & \mathrm{h} & \mathrm{i}\end{array}\right]$
$A+B=\left[\begin{array}{ccc}a+d & b+e & c+f \\ a+d & b+e & c+f \\ 2 g & 2 h & 2 i\end{array}\right]$
$|A+B|=\left|\begin{array}{ccc}a+d & b+e & c+f \\ a+d & b+e & c+f \\ 2 g & 2 h & 2 i\end{array}\right|$
$=0 \quad$ (since two rows are identical)
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