160102 Algebra (Auckland, Manawatū, and Distance) Assignment 2, 2026 | Massey University

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School of Fundamental Sciences
and School of Natural and Computational Sciences
Mathematics

160102 Algebra (Auckland, Manawatū, and Distance) Assignment 2

• 1Solve systems of linear equations and perform algebraic calculations using vectors and matrices.
• 2Use vectors to solve problems involving lines and planes in three dimensions.
• 3Calculate determinants, eigenvalues, and eigenvectors of matrices, and demonstrate the ability to use these in applications.
• 4Demonstrate proficiency with the algebra and geometry of complex numbers.
• 5Use computer software (such as MATLAB) for matrix calculations and for solving systems of linear equations.
• 6Communicate mathematical arguments in appropriate mathematical language/symbols.

Learning outcomes can change before the start of the semester you are studying the course in.

160.102  Assignment 2 Question

1. Given that z = 1 + 2i is a root of p(z) = z⁴ − 5z³ + 13z² − 19z + 10, find all roots of p(z) and use this to express p(z) as a product of (a) 4 linear factors and (b) real linear and real irreducible quadratic factors. Check your roots in Matlab using the command roots.

2. Use the substitution w = z² to find all 4 roots of z⁴ + z² + 1 and plot them in the complex plane.

3. Use de Moivre’s Theorem (e^{iθ})ⁿ = e^{inθ}, equivalently (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ), with n = 3 to obtain the triple angle formulae that express sin(3θ) and cos(3θ) in terms of sin(θ) and cos(θ).

4. Use elementary row operations (row echelon form) to compute the determinant of

[1  1  1  0
 0  2  1  0
 1  2  2  1
 1  1  1  4]

Check your answer in Matlab.

5. Show that for all square matrices A, if λ is an eigenvalue of A then λ² is an eigenvalue of A².

6. Show that for all invertible square matrices A, if λ is an eigenvalue of A then 1/λ is an eigenvalue of A⁻¹.

7. What does Matlab do when asked to provide the eigenvalues of an n × n matrix (such as

[1  1
 0  1]

) which has fewer than n linearly independent eigenvectors? How could you use Matlab to determine when a large n × n matrix has fewer than n linearly independent eigenvectors? Illustrate your method in Matlab.

8. The Matlab command randn(n) creates an n × n matrix in which each entry is a random number chosen from the normal distribution.

(a) Create such a 4 × 4 matrix and use Matlab to find its eigenvalues and eigenvectors. Use Matlab to check that the matrix of eigenvectors can be used to diagonalize the matrix. How many eigenvalues are real? If you repeat the experiment with different random numbers, do you always get the same number of real eigenvalues?

(b) Create a large such matrix (without printing it out, e.g. use A = randn(100); where the semicolon stops the result being printed out) and plot its eigenvalues as points in the complex plane. What do you notice? What happens for different values of n?

9. For large matrices, computing the eigenvalues and eigenvectors the way we have been doing it by hand is prohibitively expensive, even on a computer. There is a faster method called the power method:

Step 1. Choose any nonzero starting vector x₀.

Step 2. Let xₖ₊₁ = A xₖ for k = 0, 1, 2, … .

Step 3. Let bₖ = (xₖᵀ xₖ₊₁) / (xₖᵀ xₖ) for k = 0, 1, 2, … .

Then the sequence b₀, b₁, b₂, … tends to the eigenvalue of A of largest modulus, and xₖ tends to an eigenvector.

Let A =

[  3  -1  -1
 -12   0   5
   4  -2  -1]

and x₀ =

[1
 1
 1]

In Matlab, compute b₀, b₁, b₂, b₃, and b₄.
How do they compare to the largest eigenvalue of A?

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