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Worksheet: Representing Complex Numbers as Matrices

Q1:

Suppose we take the matrix 𝑁 =  3 4 βˆ’ 4 3  to represent the complex number 3 βˆ’ 4 𝑖 and the matrix 𝑀 =  2 βˆ’ 5 5 2  to represent the complex number 2 + 5 𝑖 .

What is the product 𝑀 𝑁 ?

  • A  6 βˆ’ 2 0 βˆ’ 2 0 6 
  • B  2 6 7 βˆ’ 7 2 6 
  • C  βˆ’ 2 0 6 6 2 0 
  • D  2 6 βˆ’ 7 7 2 6 
  • E  βˆ’ 1 4 2 3 βˆ’ 2 3 1 4 

What does it represent?

  • AThis is the matrix representing 2 6 + 7 𝑖 which is the product ( 3 βˆ’ 4 𝑖 ) ( 2 + 5 𝑖 ) .
  • BThis is the matrix representing βˆ’ 2 0 + 6 𝑖 which is the product ( 3 βˆ’ 4 𝑖 ) ( 2 + 5 𝑖 ) .
  • C This is the matrix representing 6 βˆ’ 2 0 𝑖 which is the product ( 3 βˆ’ 4 𝑖 ) ( 2 + 5 𝑖 ) .
  • DThis is the matrix representing 7 + 2 6 𝑖 which is the product ( 3 βˆ’ 4 𝑖 ) ( 2 + 5 𝑖 ) .
  • EThis is the matrix representing βˆ’ 1 4 + 2 3 𝑖 which is the product ( 3 + 4 𝑖 ) ( 2 βˆ’ 5 𝑖 ) .

Q2:

Describe the geometric transformation that occurs when numbers in the complex plane are mapped to their sum with π‘Ž + 𝑏 𝑖 .

  • Aa translation by  βˆ’ π‘Ž 𝑏 
  • Ba translation by  𝑏 π‘Ž 
  • Ca translation by  βˆ’ π‘Ž βˆ’ 𝑏 
  • Da translation by  π‘Ž 𝑏 
  • Ea translation by  βˆ’ 𝑏 βˆ’ π‘Ž 