Question

Show that if we define the complex number a + bi as the matrix: (I will attach matrix), with
normal matrix addition and multiplication, the definitions of addition and
multiplication of complex numbers are correct.

Answer 1

\[\left[\begin{matrix}a & b \\ -b & a\end{matrix}\right]\]

Answer 2

@Hunus @terenzreignz

Answer 3

@amistre64

Answer 4

@AravindG

Answer 5

an hour ago? the notifs must be lagging

Answer 6

show that a adding and multiplying it to a matrix of the same form, produces another matrix of the same form

Answer 7

\[\left[\begin{matrix}a & b \\ -b & a\end{matrix}\right]+\left[\begin{matrix}x & y \\ -y & x\end{matrix}\right]=\left[\begin{matrix}m & n \\ -n & m\end{matrix}\right]\]
\[\left[\begin{matrix}a & b \\ -b & a\end{matrix}\right]~\left[\begin{matrix}x & y \\ -y & x\end{matrix}\right]=\left[\begin{matrix}m & n \\ -n & m\end{matrix}\right] \]

Answer 8

Okay. It doesn't matter that you used m and n as the variables in your results when we originally used different variables in our computations? We're just showing that what produced is in the same form as a +bi matrix? @amistre64

Answer 9

And what can I additionally say that clarifies that the definitions of addition and multiplication of complex numbers are correct? @amistre64

Answer 10

producing the same "form" is whats important, yes

Answer 11

well, the addition of complex numbers is again a complex number ...

the multilication of complex numbers is again a complex number ...

(a+bi) + (x+yi) = (a+x) + (b+y)i

(a+bi)(x+yi) = ax + bxi + ayi -by
= (ax-by) + (ay+bx)i

Answer 12

so as long as the "form of a complex number" is maintained by the matrix operations ... all is well

Answer 13

Thanks for your help!! I appreciate it. @amistre64