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The prompt
The prompt
Mathematical inquiry processes: Verify, generalise, and prove; extend to other cases. Conceptual field of inquiry: Matrix multiplication; algebraic identities; inverse and identity matrices.
The prompt is designed for students between the ages of 16 and 18 who are studying, for example, A-level Further Mathematics or the Diploma Programme of the International Baccalaureate.
Through the inquiry, students learn to find the sum and product of matrices and become fluent in the procedures. They also learn about generalisation and proof.
Students find the prompt intriguing, particularly if they are aware that matrix multiplication is not commutative. Surely, they reason, the algebraic identity (a + b)2 = a2 + 2ab + b2 cannot hold for matrices.
The prompt is true for 2 x 2 matrices. Indeed, for a class with little or no experience of inquiry, the teacher might restrict the initial phase of the inquiry by defining A and B as 2 x 2 matrices.
The inquiry starts with students posing questions, noticing properties, and wondering about the implications of the prompt:
Why is the expansion not A2 + 2AB + B2 like it is for algebra?
The equation cannot work because when you multiply matrices AB does not equal BA.
Does it work for matrices of different sizes?
We could test 2 x 2 matrices and other square matrices.
Will it work if one of the matrices is the identity?
Will it work if one matrix is the inverse of the other?
If it works for the equation, does it work for (A - B)2?
Regulatory cards
Students use the regulatory cards to direct their inquiry. If the class is new to inquiry, the teacher invites students to answer the question "What shall we do next?" by selecting one card. In classes familiar with mathematical inquiry, students are able to construct a sequence of actions.
The matrix multiplication inquiry usually starts with students attempting to find examples (or counter-examples) in order to decide if the prompt is always true. Their results, particularly when shared and compared, lead to the making of a conjecture and a generalisation and then to proving the generalisation and analysing structure.
December 2023
Lines of inquiry
Lines of inquiry
1. Exploration
The inquiry normally begins with students testing the contention in the prompt by generating their own examples using 2 x 2 matrices.
If the students do not have prior knowledge of calculations with matrices, then they have the opportunity to ask in the initial phase. At that point the teacher's instruction, being in response to students' requests, is timely, relevant and meaningful.
Once each student in the class has tested at least one case, results can be shared and compared. The class might make a conjecture that the prompt is always true.
2. Proof
After generalising from their results, students are enthusiastic to prove the prompt is always true.
They start with the general forms of two 2 x 2 matrices and show that the two sides of the equation are the same (see the mathematical notes).
3. Change the size of the matrices
Is the prompt true if the matrices are 3 x 3?
What if the matrices are not square? Or do they have to be square to be able to find both their sum and product?
4. Increase the number of matrices
If A, B, and C are three square matrices, is it true that (A + B + C)2 = A2 + AB + AC + BA + B2 + BC + CA + CB + C2?
5. Test other identities
Do other algebraic identities work with matrices? For example, is it true that (A - B)2 = A2 - AB - BA + B2?
What about the difference of two squares?
What about cubes? Do (A + B)3 and (A - B)3 work in the same way? Is there a difference if we left or right multiply? For example, is (A + B)(A + B)2 equal to (A + B)2 (A + B)?
An alternative prompt
An alternative prompt
At an Inquiry Maths workshop at Akademeia High School in Warsaw (Poland), Dr Peter Kowalski (head of department) and Monika Klimczak-Gromska (teacher of mathematics) chose the matrix multiplication prompt to plan an inquiry collaboratively. The inquiry was for their year 12 A-level Further Maths classes.
To make connections to a concept the students had previously learnt about, Peter and Monica created a new prompt based on the binomial theorem formula. Does the binomial series expansion work if A and B are two matrices?
May 2025
Finding the inverse of a 2 x 2 matrix
Finding the inverse of a 2 x 2 matrix
Professor Merrilyn Goos (University of Queensland, Australia) carried out research into inquiry learning. She studied classes of upper secondary students, which she described as communities of inquiry.
One inquiry involved the students in discovering the formula to find the inverse of a 2 x 2 matrix. The inquiry features in the journal article Learning Mathematics in a Classroom Community of Inquiry, Journal for Research in Mathematics Education (2004).
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