# Matrix multiplication inquiry

# 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

1. Exploration

The inquiry normally begins with students testing the contention in the prompt, although classes used to generalisation and proof and fluent in finding the sum and product of matrices might try go straight to proof.

Just as with all inquiries on the website, students do not need prior knowledge to engage with the prompt. If they do not know how to carry out 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. Proving

After generalising from their results, students are enthusiastic to prove the prompt must always be true.

Each element of the matrices becomes a variable in the general case. A valuable discussion often occurs when the teacher requires a strict definition of the variables in terms of sets of numbers.

Students apply the same techniques from the particular case to show that the left- and right-hand sides are equal in all cases (see the mathematical notes).

3. Changing the size of the matrices

Is the prompt true if the matrices are 3 x 3?

What if they are not square? Or do they have to be square to be able to find both their sum and product?

4. Increasing the number of matrices

Is it true that (A + B + C)2 = A2 + AB + AC + BA + B2 + BC + CA + CB + C2?

5. Testing 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)?

# Finding the inverse of a 2 x 2 matrix

Professor Merrilyn Goos (of the University of Queensland) 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 discovering the algorithm for finding 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).