Julia Goedecke

A combination of tasks: checking which matrix products exist, calculating some of these products, calculating transpose matrices. Comparing product of transpose with transpose of product. Experiencing associativity of matrix multiplication. Not much randomisation, only in which matrix product is computed as second option.

Comprehensive solution written out in Advice.

Use matrix multiplication to get an equation for \(k\) which is then to be solved.

To understand matrix multiplication in terms of linear combinations of column vectors.

Asking the student to create examples of two matrices which multiply to zero but are not themselves the zero matrix. Then getting the student to think about some features of these examples.

Easy intro questions to be done when the students have seen the "vector space axioms" but not as axioms, just in the context of \(\mathbb{R}^n\).

Calculating with vectors in \(\mathbb{R}^4\), including resolving brackets. The fixed vector size is so that a test is fair to all students.

The student is asked to write a matrix with a certain property, or tick a box labelled "this is impossible" if it can't be done.

A custom marking algorithm on the gap-fill part first checks if the student ticked the box. If they did, their answer is marked correct if it really is impossible. If they didn't tick it, their matrix is checked against the required property.

Product of one of 2, 3, 5, 9, or 10 by a number up to 10. With hints to learn calculation rather than memorisation.

This is supposed to demonstrate allowing one of two different free variables in the student's answer, but only marked as correct if the same free variable is used in all gaps. The custom marking algorithm should extend to any number of gaps, and one could add more alternative answers to allow for more free variable names. It doesn't allow just any free variable name.