Today we see another example of how matrices are used in mathematics. Students are given a problem to solve. Most students think the way to solve this problem is by dividing by the 2X2 matrix. When they check their answer on the calculator they get an error. When students ask I say "What does dividing by 3 in 3x=12 is the same as doing what? How do we get undo a function like a square root when we solve?" Some students will think about the inverse while others will not be sure what to do.
If students find the correct answer I have then put the answer on the board and then explain how they did the problem. If there are no correct answers I will start developing the process of solving a matrix equation.
During the Bell Work, students came to the realization that solving a matrix equation is not a straightforward process of applying well known strategies. With this fresh in their minds, I begin developing the process of using inverses to solve.
I display a slide that has several different problems. The first problem is an algebraic problem. Students realize they need to square both sides to find the answer. "Why do we square both sides?" I want students to comment on using the inverse of the square root.
I now go back to the 3x=12 and ask how they do this problem. Students say divide by 3. I ask,"Why do you divide by 3?" Many students say to make the 3 become a 1. I respond, "the problem says 3 times something is 12. So when you divide by 3 what are you doing?" I hope that some students will follow the pattern from the first example and begin to see that the inverse operation is being used.
I now move to AX=B and the matrix problem. I ask, "When you divided by the 2X2 matrix on the calculator, what was the result?" When students reply that an error message appeared I will ask, "What could we do instead of dividing by the matrix, since that doesn't work?" We will think about this for a while until students begin to realize that another possibility is to multiply by the inverse of the matrix. In the example, AX = B, the inverse is A^(-1).
It is now important to talk about order when we multiply by the inverse. I ask, "Does AB = BA for all matrices?" From our Do matrices work like real numbers? lesson students know that the answer is "No." Building on this, I have the students look at the equation with the matrices. I say, "We need to find the inverse of the 2X2 matrix. What will the dimension of the inverse be?" Students know that the inverse will be a 2X2. "Where should the inverse be places before or after the 2X1 matrix?" After some thought and discussion about which position will give us the correct answer students determine the inverse should be in front of the 1X2 matrix.
We write the notation in the middle expression and then I ask the last equation, "Do you notice how the inverse is in the same position on both sides of the equation?" I ask students what the result is when we multiply the inverse matrix by the original matrix. This reminds students about the identity matrix which is important for precision. If I just write the matrix with x and y students do not really understand what is happening we don't end up with 1 but a matrix that works similar to 1. I now ask, "if you multiply a matrix by the identity matrix what should you get?" Students realize the original matrix is the answer to the right side of the equation. For now I have the students use the calculator to compute the result. I do this to save time.
I now ask, "What are the final values for x and y?"
After working with this problem students get into groups and solve a practice problem. Students are given several minutes and the values of x, y and z are put on the board when most groups are finished.
I now return to the original Bell Problem but ask students to multiply the right side together. As students multiply they begin to see how the right side represents an expression with an x and a y. Once the multiplication is complete I ask, "What is the dimension of the matrix on the right? Left? Since this is an equation we know the matrices are equal. What is true about matrices that are equal?" When students say the terms must be equal, I write out what this means (2x+6y=-1 and x+8y=2) The students now see that we have a system of equations. "How can we use matrices to solve a system of linear equations?" We discuss where to place the coefficients of each term and where the x and y are positioned. After discussing how to organize the matrices students explain the process of solving.
I give students two systems to solve. Students work for several minutes on each problems. After about 4-5 minutes I randomly pick students to share the solution.
As class ends students need to develop notes to use as they practice solving. I have students find their Summary of operations notes. Students are told to add using matrices to solve systems and write an explanation on the process. I suggest writing an example may help them remember the process.
I also give students Problem Solving Using Systems of Linear Equations. This worksheet has problems that are common on standardized tests (mixture problems and interest problems). The students must find the equations and then use their equations to solve problems.