Let alpha=a+bi and beta=c+di be complex scalars and let A and B be matrices with complex entries.

BenoguigoliB

BenoguigoliB

Answered question

2021-09-16

Let α=a+bi  and  β=c+di be complex scalars and let A and B be matrices with complex entries.
(a) Show that
α+β=α+β  and  αβ=αβ
(b) Show that the (i,j) entries ofAB and A¯B¯ are equal and hence that
AB=AB

Answer & Explanation

Brittany Patton

Brittany Patton

Skilled2021-09-17Added 100 answers

Step 1
(a)
Let α=a+bi  and  β=c+di
α+β=(a+bi)+(c+di)
=(a+c)+i(b+d)
=(a+c)i(b+d)
=a+cibid
=(aib)+(cid)
=α+β
Similarly,
αβ=(a+bi)(c+di)
=acbd+i(bc+ad)
=acbdi(bc+ad)
=(aib)(cid)
=αβ
Thus, α+β=α+β  and  αβ=αβ
Step 2
(b)
Let A nad B are the matrices whose entries are from complex numbers. (i,j)th entry of AB is ai1b1j+ai2b2j+s˙+abnj
From part (a),
ai1b1j+ai2b2j+s˙+abnj=ai1b1j+ai2b2j+s˙+abnj
=ai1b1j+ai2b2j+s˙+abnj
This implies that i,j) th entry of AB  and  AB are same.
It is known that if each of entry of two matrices are same then they are equal.
Hence, AB=AB

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