At universities, colleges and other educational institutions, students are constantly sweating it over their exams. The Engineer wondered: what makes you get out of it? Here you’ll find the answers to the latest question featured in the magazine.

The subject

The following questions are taken from an exam for the first-year module ‘Linear Algebra’ on the Bachelor’s programme in Circular Engineering at Maastricht University. We received them from the course co-ordinator, mathematical engineer Martijn Boussé.

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 include true/false questions and construction questions, for which students need to have a clear grasp of the definitions.

The questions

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

a) A 123456789 × 123456789 matrix in which all elements are equal to 1/123456789 has only one eigenvalue that is not equal to 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 diagonalised and invertible matrix with orthogonal columns, all of whose elements are non-zero.

The answers

1) True. The matrix has an eigenvalue of zero because its columns are linearly dependent, and it has an eigenvalue of 1 because all its rows sum to 1.

The nullity of the matrix is 123456789−1 because there is only one pivot, so the multiplicity of the eigenvalue 0 is 123456789 − 1. The only other eigenvalue is 1.

2) False. Counterexample: the vector v = [2 1]^T lies in both the column space and the null space of the matrix A = [-2 4; -1 2]. Indeed, Av = 0 and v = -1*a1 + 0*a2.

3) For example, [-2 1; 1 2]. This matrix is symmetric and therefore (orthogonally) diagonalizable according to the spectral theorem. It has no proportional columns and is therefore invertible. Furthermore, the inner product of the columns is zero, so the columns are orthogonal.