If A= [ cc 1 & 2 i -3 & 2 ] and B= [ cc 2 i & 1 2 & -3 ] where i=-1, find A+B and A-B. Show that A + B is singular. Is A – B singular? Justify your answer.

A+B & = [ cc i & 2 i -3 & 2 ]+ [ cc 2 i & i 2 & -3 ] & = [ cc i+2 i & 2 i+i -3+2 & 2-3 ]= [ cc 3 i & 3 i -1 & -1 ] |A+B|= | cc 3 i & 3 i -1 & -1 | =3 i(-1)-(-1) 3 i=0 A+B is a singular matrix. A-B & = [ cc i & 2 i -3 & 2 ]- [ cc 2 i & i 2 & -3 ] & = [ cc i-2 i & 2 i-i -3-2 & 2+3 ]= [ cc -i & i -5 & 5 ] |A-B|= | ll -i & i -5 & 5 |=(-i) 5-(-5) i=0 A-B is also a singular matrix.

If A= [ cc 1 & 2 i -3 & 2 ] and B= [ cc 2 i & 1 2 & -3 ] where i=…

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If $\mathrm{A}=\left[\begin{array}{cc}1 & 2 i \\ -3 & 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}2 i & 1 \\ 2 & -3\end{array}\right]$ where $i=\sqrt{-1}$, find $\mathrm{A}+\mathrm{B}$ and $\mathrm{A}-\mathrm{B}$. Show that $\mathrm{A}$

+ B is singular. Is A – B singular? Justify your answer.

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Answer

$\begin{aligned} A+B & =\left[\begin{array}{cc}i & 2 i \\ -3 & 2\end{array}\right]+\left[\begin{array}{cc}2 i & i \\ 2 & -3\end{array}\right] \\ & =\left[\begin{array}{cc}i+2 i & 2 i+i \\ -3+2 & 2-3\end{array}\right]=\left[\begin{array}{cc}3 i & 3 i \\ -1 & -1\end{array}\right]\end{aligned}$

$\therefore \quad|A+B|=\left|\begin{array}{cc}3 \mathrm{i} & 3 \mathrm{i} \\ -1 & -1\end{array}\right|$

$=3 \mathrm{i}(-1)-(-1) 3 \mathrm{i}=0$

$\therefore \quad \mathrm{A}+\mathrm{B}$ is a singular matrix.

$\begin{aligned} A-B & =\left[\begin{array}{cc}i & 2 i \\ -3 & 2\end{array}\right]-\left[\begin{array}{cc}2 i & i \\ 2 & -3\end{array}\right] \\ & =\left[\begin{array}{cc}i-2 i & 2 i-i \\ -3-2 & 2+3\end{array}\right]=\left[\begin{array}{cc}-i & i \\ -5 & 5\end{array}\right]\end{aligned}$

$\therefore \quad|\mathrm{A}-\mathrm{B}|=\left|\begin{array}{ll}-\mathrm{i} & \mathrm{i} \\ -5 & 5\end{array}\right|=(-\mathrm{i}) 5-(-5) \mathrm{i}=0$

$\therefore \quad \mathrm{A}-\mathrm{B}$ is also a singular matrix.

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