The following questions are taken from an exam for the first-year module ‘Linear Algebra’ on the Bachelor’s programme Circular Engineering programme at Maastricht University. De Ingenieur received these questions from the course co-ordinator and mathematical engineer Martijn Boussé (see also the article on page 32). The answers will be available from 22 September here .

Edited by Marlies ter Voorde

Matrices and vectors are widely used in linear algebra. These are fundamental mathematical structures for organising numbers, performing mathematical operations and discovering relationships – the latter based on certain properties of the matrices and vectors that can be calculated.

Typical exam questions in linear algebra are true/false questions and construction questions, which require students to have a firm grasp of the definitions. You are welcome to use Google.

The exam question

1 Are the following statements true or false? And why?

a A 123456789 × 123456789 matrix, all elements of which are equal to 1/123456789, has only one eigenvalue that is not zero, and that is equal to 1.

b There are no non-zero vectors that lie in both the null space and the column space of a matrix.

2 Construct a diagonaliseable and invertible matrix with orthogonal columns, all of whose elements are non-zero.