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?

  • A 2 6 7 7 2 6
  • B 2 6 7 7 2 6
  • C 2 0 6 6 2 0
  • D 6 2 0 2 0 6
  • E 1 4 2 3 2 3 1 4

What does it represent?

  • AThis is the matrix representing20+6𝑖 which is the product (34𝑖)(2+5𝑖).
  • B This is the matrix representing 620𝑖 which is the product (34𝑖)(2+5𝑖).
  • CThis is the matrix representing 14+23𝑖 which is the product (3+4𝑖)(25𝑖).
  • DThis is the matrix representing 7+26𝑖 which is the product (34𝑖)(2+5𝑖).
  • EThis is the matrix representing 26+7𝑖 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 𝑏𝑎
  • Ca translation by 𝑎𝑏
  • 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.

  • A 𝑀
  • B 𝑀 T
  • C 𝑀 T
  • D 𝑀

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 𝑧.
  • BThe matrix inverse, 𝑀, corresponds to the complex number 𝑧.
  • CThe matrix inverse, 𝑀, corresponds to the complex number 1𝑧.
  • 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?

  • A 6 1 1 6
  • B 6 1 5 1 5 6
  • C 4 1 5 1 5 4
  • D 6 1 1 6
  • E 4 1 5 1 5 4

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

  • A 𝑧 + 𝑧 = 6 𝑖
  • B 𝑧 𝑧 = 6 𝑖
  • C 𝑧 𝑧

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.

  • A 1 3 3 1
  • B 1 1 0 1 3 3 1
  • C 1 1 0 1 3 3 1
  • D 1 3 3 1
  • E 3 9 3 7 3 7 3 9

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.

  • A 𝑀 + 𝑀 𝑀
  • B 𝑀 + 𝑀 𝑀 T
  • C 𝑀 + 𝑀 𝑀 T
  • D 𝑀 + 𝑀 𝑀 T
  • E 𝑀 + 𝑀 𝑀 T

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?

  • A 1 𝑧 = 𝑧 | 𝑧 |
  • B 𝑧 = 1 𝑧 | 𝑧 |
  • C 1 𝑧 = 𝑧 | 𝑧 |
  • D 𝑧 = 1 𝑧 | 𝑧 |
  • E 𝑧 = 1 𝑧 | 𝑧 |

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