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Complex numbers and matrices

Written by Hassan Raza (Mathematics)

ECAT · Mathematics · 8 questions

Name: ____________________   Date: ____________

  1. What is i⁷, where i is the square root of -1?

    • A -i
    • B i
    • C -1
    • D 1

  2. What is the modulus of the complex number 3 + 4i?

    • A 7
    • B 25
    • C 1
    • D 5

  3. What is (2 + 3i)(2 - 3i)?

    • A -5
    • B 4 - 9i
    • C 13
    • D -13

  4. What is the determinant of the matrix [[2, 3], [1, 4]]?

    • A 5
    • B 11
    • C 8
    • D -5

  5. What is the complex conjugate of 5 - 2i?

    • A 2 - 5i
    • B -5 + 2i
    • C -5 - 2i
    • D 5 + 2i

  6. The matrix [[1, 2], [2, 4]] is:

    • A Singular
    • B Invertible
    • C The identity matrix
    • D A zero matrix

  7. What is (1 + i)²?

    • A 2i
    • B 1 + 2i
    • C 0
    • D 2

  8. If I is the 2 x 2 identity matrix and A is any 2 x 2 matrix, then I A equals:

    • A The zero matrix
    • B A transposed
    • C The inverse of A
    • D A

Answer key

1. A, 2. D, 3. C, 4. A, 5. D, 6. A, 7. A, 8. D

  1. i⁴ = 1, so i⁷ = i⁴ x i³ = i³ = -i.

  2. The modulus is sqrt(3² + 4²) = 5.

  3. (a + bi)(a - bi) = a² + b² = 4 + 9 = 13.

  4. The determinant is 2 x 4 - 3 x 1 = 5.

  5. The conjugate changes the sign of the imaginary part, giving 5 + 2i.

  6. Its determinant is 1 x 4 - 2 x 2 = 0, so it has no inverse and is singular.

  7. (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i.

  8. Multiplying by the identity leaves a matrix unchanged.