The Matrix
Problem
A Matrix is a rectangular array of numbers (which are called the entries of the matrix). The plural of matrix is matrices.
The matrix $
\begin{pmatrix}2&-3&\phantom{-}5\\0&\phantom{-}8&-1\end{pmatrix} $ has two rows and three columns. We say that the dimension of the matrix is $2 \times 3$.
You can add and multiply two matrices if they satisfy certain conditions. To add two matrices they must have the same dimensions, and then you can add together the corresponding elements.
Example:
$$
\begin{pmatrix}\phantom{-}2&\phantom{-}0\\\phantom{-}9&-3\\\phantom{-}1&\phantom{-}4\end{pmatrix}+\begin{pmatrix}-5&\phantom{-}1\\\phantom{-}2&\phantom{-}3\\\phantom{-}0&-1\end{pmatrix}=\begin{pmatrix}2+
-5&\phantom{-}0+1\\9+2&-3+3\\1+0&\phantom{-}4+-1\end{pmatrix}=\begin{pmatrix}-3&\phantom{-}1\\\,11&\phantom{-}0\\\phantom{-}1&\phantom{-}3\end{pmatrix}
$$
To multiply two matrices together then the number of columns of the first matrix has to be equal to the number of rows of the second matrix. The example below shows how one element of the product of two matrices has been calculated - can you see how the other elements have been generated?
You can use this matrix to check how the other elements were calculated. If you click on entries in the answer matrix you will see the relevant rows and columns highlighted and the calculations used to generate the entry.
Calculate these matrix products:
- $\begin{pmatrix}3&-3\\2&0\\1&4\end{pmatrix} \begin{pmatrix}2&-1&5\\0&3&-2\end{pmatrix}$
- $\begin{pmatrix}2&-1\\3&5\end{pmatrix}\begin{pmatrix}5&-3\\-1&0\end{pmatrix}$
- Let ${\bf P}=\begin{pmatrix}2&3&-1\end{pmatrix}$ and let ${\bf Q}=\begin{pmatrix}-1\\0\\5\end{pmatrix}$. Find the products ${\bf P}{\bf Q}$ and ${\bf Q}{\bf P}$.
- Let ${\bf A} = \begin{pmatrix}3&-1&0\\-2&5&1\end{pmatrix}$ and ${\bf B}=\begin{pmatrix}3&2 \\ 0 & -1\end{pmatrix}$. State which of the products ${\bf A}{\bf B}$ and ${\bf B}{\bf A}$ can be calculated and find this product.
You might like to use this Matrix Multiplication Calculator to check your answers. If you click on an entry in the answer matrix you can see how the value was calculated.
For the rest of this question we will consider square $2 \times 2$ matrices, so matrices of the form $\begin{pmatrix}a&b\\c&d\end{pmatrix}$. Here are some questions to think about:
- With numbers it does not matter which order we use to multiply, so $2 \times 3 = 3 \times 2$. Is the same true of matrices, i.e. do we have ${\bf A}{\bf B} = {\bf B}{\bf A}$ in all cases?
- If ${\bf A}{\bf B}={\bf 0}$, must we have at least one of ${\bf A}$ or ${\bf B}$ equal to ${\bf 0}=\begin{pmatrix}0&0\\0&0\end{pmatrix}$?
- Consider the matrix ${\bf M}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$. What is the matrix ${\bf M}^{2023}$?
- The $2 \times 2$ matrix ${\bf P}$ satisfies ${\bf P}{\bf X} = {\bf X}{\bf P}$ for all possible $2 \times 2$ matrices ${\bf X}$. Find the possible matrices that ${\bf P}$ could be.
You could write ${\bf P} = \begin{pmatrix}a&b\\c&d\end{pmatrix}$. The equation works for all possible ${\bf X}$, so you can try using a simple specific matrix ${\bf X}$ and see what you can deduce about ${\bf P}$. Then try another matrix ${\bf X}$ to see if you can deduce something else!
There are more matrix problems in this feature.
Getting Started
Student Solutions
Calculate these matrix products:
1. $\begin{pmatrix}3&-3\\2&0\\1&4\end{pmatrix} \begin{pmatrix}2&-1&5\\0&3&-2\end{pmatrix}$
Tanish from Pate's Grammar School in the UK, Ci Hui from Queensland Academy for Science Mathematics and Technology (QASMT) in Australia, Mizuki and Yuhan from St George's British International School, Rome in Italy and Beren from the United Kingdom correctly found this matrix product. This is Ci Hui's work:
2. $\begin{pmatrix}2&-1\\3&5\end{pmatrix}\begin{pmatrix}5&-3\\-1&0\end{pmatrix}$
Tanish, Mizuki, Yuhan, Ci Hui and Beren correctly found this matrix product. Here is Beren's work:
3. Let ${\bf P}=\begin{pmatrix}2&3&-1\end{pmatrix}$ and let ${\bf Q}=\begin{pmatrix}-1\\0\\5\end{pmatrix}$. Find the products ${\bf P}{\bf Q}$ and ${\bf Q}{\bf P}$.
Tanish, Mizuki, Yuhan, Beren and Ci Hui worked out both products. Here is Mizuki's work:
4. Let ${\bf A} = \begin{pmatrix}3&-1&0\\-2&5&1\end{pmatrix}$ and ${\bf B}=\begin{pmatrix}3&2 \\ 0 & -1\end{pmatrix}$. State which of the products ${\bf A}{\bf B}$ and ${\bf B}{\bf A}$ can be calculated and find this product.
Tanish, Mizuki, Yuhan, Beren and Ci Hui all concluded that only one product could be calculated. This is Tanish's work:
For the rest of this question we will consider square $2 \times 2$ matrices, so matrices of the form $\begin{pmatrix}a&b\\c&d\end{pmatrix}$. Here are some questions to think about:
1. With numbers it does not matter which order we use to multiply, so $2 \times 3 = 3 \times 2$. Is the same true of matrices, i.e. do we have ${\bf A}{\bf B} = {\bf B}{\bf A}$ in all cases?
Beren explained why this may not be true of matrices:
This is not true. With matrices the answer will change as the multiplication is different.
Ci Hui used an example to show that this is not true for all matrices:
Mizuki and Yuhan also found examples to show that it is not true for all matrices. Mizuki and Yuhan's examples are of square matrices. This is Yuhan's work, which also includes an example of two matrices $\bf A$ and $\bf B$ where ${\bf A}{\bf B}$ is equal to ${\bf B}{\bf A}.$
Tanish used algebra to show that, in general, ${\bf A}{\bf B} \neq {\bf B}{\bf A}:$
2. If ${\bf A}{\bf B}={\bf 0}$, must we have at least one of ${\bf A}$ or ${\bf B}$ equal to ${\bf 0}=\begin{pmatrix}0&0\\0&0\end{pmatrix}$?
Beren explained how ${\bf A}{\bf B}$ could be ${\bf 0}$ even if ${\bf A}$ and ${\bf B}$ are not equal to ${\bf 0}:
This is not true because we are multiplying them so each position could be perhaps $1-1$ which will total to $0.$
Mizuki and Ci Hui both found pairs of matrices where neither matrix is the zero matrix, but the product is the zero matrix. This is Mizuki's example:
This is Ci Hui's example:
Tanish used algebra to describe two families of matrices where ${\bf A}$ and ${\bf B} are not ${\bf 0},$ but ${\bf A}{\bf B}={\bf 0}:$
3. Consider the matrix ${\bf M}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$. What is the matrix ${\bf M}^{2023}$?
Tanish, Yuhan, Ci Hui and Mizuki worked this out by calculating the first few powers of ${\bf M}.$ This is Yuhan's work:
4. The $2 \times 2$ matrix ${\bf P}$ satisfies ${\bf P}{\bf X} = {\bf X}{\bf P}$ for all possible $2 \times 2$ matrices ${\bf X}$. Find the possible matrices that ${\bf P}$ could be.
Mizuki found one matrix that ${\bf P}$ could be:
Ci Hui and Tanish noticed that ${\bf P}$ could be the identity matrix, and Tanish found a family of matrices that ${\bf P}$ could be:
In fact, there are even more matrices that satisfy ${\bf P}{\bf X} = {\bf X}{\bf P}$ for all possible $2 \times 2$ matrices ${\bf X}.$ Can you find any more?
Teachers' Resources
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.
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.