Worksheet: Matrix Representation of Complex Numbers

In this worksheet, we will practice representing a complex number as a linear transformation matrix and using this to determine the product of two complex numbers.

Q1:

Suppose we take the matrix 𝑁=3443 to represent the complex number 34𝑖 and the matrix 𝑀=2552 to represent the complex number 2+5𝑖.

What is the product 𝑀𝑁?

  • A206620
  • B620206
  • C14232314
  • D267726
  • E267726

What does it represent?

  • AThis is the matrix representing 14+23𝑖 which is the product (3+4𝑖)(25𝑖).
  • BThis is the matrix representing20+6𝑖 which is the product (34𝑖)(2+5𝑖).
  • CThis is the matrix representing 26+7𝑖 which is the product (34𝑖)(2+5𝑖).
  • DThis is the matrix representing 620𝑖 which is the product (34𝑖)(2+5𝑖).
  • EThis is the matrix representing 7+26𝑖 which is the product (34𝑖)(2+5𝑖).

Q2:

Describe the geometric transformation that occurs when numbers in the complex plane are mapped to their sum with 𝑎+𝑏𝑖.

  • Aa translation by 𝑏𝑎
  • Ba translation by 𝑎𝑏
  • Ca translation by 𝑏𝑎
  • Da translation by 𝑎𝑏
  • Ea translation by 𝑎𝑏

Q3:

Let 𝑀 be the matrix representation of the complex number 𝑧=𝑎+𝑏𝑖.

Express the matrix representation of 𝑧 in terms of 𝑀.

  • A𝑀T
  • B𝑀T
  • C𝑀
  • D𝑀

What does det𝑀 correspond to in relation to 𝑧?

  • AThe determinant, det𝑀, corresponds to |𝑧|.
  • BThe determinant, det𝑀, corresponds to |𝑧|.
  • CThe determinant, det𝑀, corresponds to |𝑧|.
  • DThe determinant, det𝑀, corresponds to 𝑧.

What does 𝑀 correspond to in terms of 𝑧?

  • AThe matrix inverse, 𝑀, corresponds to the complex number 1𝑧.
  • BThe matrix inverse, 𝑀, corresponds to the complex number 𝑧.
  • CThe matrix inverse, 𝑀, corresponds to the complex number |𝑧|.
  • DThe matrix inverse, 𝑀, corresponds to the complex number 𝑧.

Q4:

Suppose we take the matrix 𝑀=5775 to represent the complex number 𝑧=5+7𝑖 and the matrix 𝑁=1881 to represent the complex number 𝑧=18𝑖.

What is the sum 𝑀+𝑁?

  • A6116
  • B415154
  • C615156
  • D6116
  • E415154

What does this represent in terms of the complex numbers 𝑧 and 𝑧?

  • A𝑧𝑧=6𝑖
  • B𝑧𝑧
  • C𝑧+𝑧=6𝑖

Q5:

Let 𝑀=711117 be the matrix representation of the complex number 𝑧 and 𝑀=4114 be the matrix representation of 𝑧. Determine the matrix representation of 𝑧𝑧.

  • A1101331
  • B1331
  • C1331
  • D1101331
  • E39373739

Q6:

Let the complex numbers 𝑧, 𝑧, and 𝑧 be represented by the matrices 𝑀, 𝑀, and 𝑀. Rewrite the following complex calculation as a matrix calculation: (𝑧+𝑧)𝑧.

  • A𝑀+𝑀𝑀T
  • B𝑀+𝑀𝑀T
  • C𝑀+𝑀𝑀T
  • D𝑀+𝑀𝑀
  • E𝑀+𝑀𝑀T

Q7:

Let 𝑀 be the matrix representation of the complex number 𝑧=𝑎+𝑏𝑖. What is the corresponding complex number identity for the matrix identity 𝑀=1𝑀𝐶,detT where 𝐶 is the cofactor matrix of 𝑀?

  • A1𝑧=𝑧|𝑧|
  • B1𝑧=𝑧|𝑧|
  • C𝑧=1𝑧|𝑧|
  • D𝑧=1𝑧|𝑧|
  • E𝑧=1𝑧|𝑧|

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