Yesterday our focus was converting a system of equations to a matrix equation. Today we will focus on solving the matrix equation. There will have to be a certain amount of direct instruction for this process, so I felt the best way to convey this information is through a PowerPoint.
I start by getting students to recap what we worked on yesterday. Mainly that we tried to figure out how many cat toys we can make with the available catnip and stuffing and that we could not divide both sides of our equation by a matrix.
Slide #3 gets students thinking about how we can solve our matrix equation without division. I use a regular equation as an analogy and we will use that line of thinking to apply to matrices. I discuss more about my teaching strategies of this portion of the lesson in the video below.
Slides #5-7 introduce students to identity matrices and inverse matrices, and I give information about both. I don’t expect students to be able to come up with the formula for the inverse on their own, so I instead to focus instead on verifying that the inverse times the original matrix actually gives the identity matrix.
After presenting the definition of the inverse matrix, I will give them some time to work on this with their table groups and I will instruct them to see if they can finish up the problem and solve for X. You may have to remind them to check that their answer is the same as the solution from yesterday. I will choose a student to share their work with the class after a few minutes.
On the right side of the matrix equation, my students will often try to multiply BA-1 instead of A-1B. Asking them to consider the dimensions of the two matrices is usually enough for them to catch their mistake and realize that they do not match up and must be reversed. If no one does this, I will still bring it up and ask students if it is okay and see what they say.
After finishing up this first example, I want to move on to an example that has three variables instead of two. Students won’t realize the power of this method until they see it with an example that they could not solve algebraically. On slide #9 of the PowerPoint there is such a question – now we have three variables and I want to see if they can solve using matrices. I have them work in their table groups to get as far as they can.
They will probably realize that we don’t know how to find the inverse of a 3 by 3 matrix. At this point I will show them how to find it on their calculator. This is the most efficient way, so I will not spend too much time explaining the process with row operations. However, if there is time I will explain the process in general so that students are exposed to the process.
Finally, the last slide has students summarize the process of solving a system using a matrix. This is a good time for students to summarize the work we did over the last two days. We went through a lot of new concepts, so this is a good time for you to go around and look at their summaries to make sure that students are to the point you want them to be. It is also good to keep an eye and ear out for the new vocabulary to make sure that they understand it and are using it correctly.