The Matrix

This problem introduces matrix multiplication.

There are some numerical examples to start with, followed by some questions about $2 \times 2$ matrices, which explore the similarities and differences between real number multiplication and matrix multiplication.

For the second set of problems, students might like to consider what the corresponding result would be if it was two real numbers being multiplied (for example, if $ab=0$ then we must have $a$ or $b$ - or both - equal to 0).  They could also consider how the results might change if the matrices had a dimension other than $2 \times 2$.

Possible Support

The following matrix calculators might be helpful for students who are finding multiplication tricky, or to check their answers to the first 4 questions.  They can also be used to test out ideas when considering the second set of 4 questions.

Matrix multiplication calculator - clicking on a cell in the resultant matrix shows the calculation used to find the value which might be useful for students who are finding multiplication tricky.

Matrix power calculator

Possible extension

If we consider the sequence $a, a^2, a^3, \ldots$ where $a$ is a real number, then the sequence either diverges (if $|a| >1$), converges (if $|a| <1$), stays constant (if $a=1$) or is periodic with period 2 (if $a=-1$).

If ${\bf M}$ is a $2 \times 2$ matrix, what sorts of behaviour can the sequence ${\bf M}, {\bf M}^2, {\bf M}^3, \ldots$ have?

There are more matrix problems in this feature.