## Checkpoint - Course 2, Unit 1 Matrix Models

In each unit, there is a final lesson and Checkpoint that helps students summarize the key ideas in the unit. The final Checkpoint will generally be discussed in class, with the teacher facilitating the summarizing, and students making notes in their Math Toolkits (teachers may just refer to this as "notes") of any points they need to remember, adding illustrative examples as needed. If your student is having difficulty with any investigation in this unit, this Checkpoint and the accompanying answers may help you recall the concepts involved, and give you the big picture of what the entire unit is about. If your student has completed the unit, then a version of this should be in his or her notes or toolkit. Students should also have Technology Tips in their toolkits, which may be useful for this unit.

Possible Responses to Unit Summary Checkpoint
In this unit, students used matrix models to analyze problem situations and they examined properties of matrices. The bolded words are vocabulary and concepts your student should be familiar with.
a. In order for information to be useful it must be organized.
• Describe how matrices can be used to organize information. Matrices are rectangular arrays with rows and columns of numbers that can be used to organize information about two variables. For example, consider average running time for boys and girls of different ages. The values of one variable, like gender, are the labels for the rows, and the values of the other variable, like age, are the labels for the columns. Then the individual cells or entries in the matrix give the average running time for a particular gender at a particular age. Once the information has been entered into a matrix, it may be appropriate to operate on the matrix, by multiplying, for example, to elicit more information.
In general, any model has two purposes: to elucidate the mathematics of the current situation; and to permit operations that will produce new information. Students will learn about statistical models, function models, parametric models, graph models, and many others in CMIC. For example, if a coordinate graph models the relationship between dependent and independent variables, say these are height (h) and time (t), then replacing t with t - 3 produces a new graph with new information about values of height if the times are delayed by 3 units. Or, if a matrix models sales of three models of cars in a current year, then multiplying by 0.1 will give a new matrix predicting sales if there is an increase of 10% across the board.
• Can the same information be displayed in a matrix in different ways? Explain. Yes, you might switch the rows and columns, or list the rows and columns of a matrix in different orders. This would not alter the information at all. The dimensions would be reversed.
• What are some advantages of using matrices to organize and display information? What are some disadvantages? Matrices are an extension of tables, usable for two-variable data. They are easily stored in and operated on by computers, allowing quick and convenient analysis of the data. Possible disadvantages include that only two variables can be included in a matrix model. In addition, matrices are not as visual as graphs; this means that numerical patterns in the data are not so easy to spot in a matrix format, and also that when using matrices to solve systems of linear equations care must be taken with solutions to interpret the results. The matrix solution of a system only works when a matrix has an inverse, but there are two occasions when a matrix has no inverse; one indicates many solutions, the other no solution. A graphical solution makes this distinction obvious; a matrix solution does not.
In Course 3, students will extend matrix models to solving linear programming problems; in this case, all intersection points are not helpful in the solution, a problem that is clear in a graphical solution, but not a matrix solution. In general, it is important that students develop an awareness of different solution strategies and are able to weigh the pros and cons of each.
b. Sometimes a situation involves two variables that are linked by two or more conditions. These situations can often be modeled by a system of two linear equations of the format ax + by = c. Describe at least three different methods for solving such a system of linear equations.
• In Lesson 3, students learned how to set up a matrix equation of the form Ax = B, where A and B are matrices, and then solve by multiplying each side by the inverse of matrix A, probably using a calculator to find the inverse matrix. For example, if 2x + 3y = 5 and 3x - 6y = 11, then the matrix equation is  =  or AX = B. The solution is X = A-1B = .
• Students could also rewrite the equations in the format "y = ax + b" so that they can both be graphed on a calculator, to find the intersection point. This method is one students used in Course 1. For the above example, this would mean graphing y = 5/3 - (2/3)x and y = (1/2)x - 11/6.
• Again relying on Course 1 methods, students could make tables by hand of the equations in their current format, substituting values for x and finding corresponding values for y, and then graphing these by hand to find the intersection. In the above example, the tables might look like:

 x y 1 1 2 1/3 3 -1/3 4 -1
 x y 1 -8/6 2 -5/6 3 -2/6 4 1/6

• They might make tables by hand or calculator and use the tables to find the solution.
• They might rewrite the equations in the format "y = ax + b" and then substitute to make an equation with only one variable, which can then be solved using algebraic rules for operating on equations. In the above example, we would have 5/3 - (2/3)x = (1/2)x - 11/6, which would then become 21/6 = (7/6)x, so x must be 3. This is a skill students learned to use in Course 1.
c.

List all the different operations on matrices that you have investigated in this unit.
Students learned, in Lesson 1, to use row or column sums to give total sales of a particular brand of shoes or in a particular month (see Investigation 1), to multiply a matrix by a number to predict the auto production if there is a 10% increase (see Investigation 3), and to add or subtract two matrices to compare auto production (see Investigation 3). In Lesson 2, students learned to multiply two matrices to find the costs of ordering uniforms from two different companies for three different teams (see Investigation 2). Still in Lesson 2, students extended multiplication of matrices to finding powers, and used these results to analyze a food web (see Investigation 3).

d. For each operation listed in Part c:
• Describe how to use the operation using paper and pencil. To add (or subtract) matrices, you add (or subtract) corresponding entries. To multiply by a number, you multiply every entry of the matrix by that number. To multiply two matrices A and B, you multiply rows of matrix A by columns of matrix B, as follows: you multiply the first entry of row 1 matrix A by the first entry of column 1 matrix B, then the second entry of row 1 matrix A by the second entry of column 1 matrix B, and so on until all row 1 and column 1 entries have been multiplied. Then these are summed to make the (row 1, column 1) entry of AB. Now multiply row 1 matrix A by column 2 matrix B in the same way to get the (row 1, column 2) entry of AB. Proceed doing this until row 1 matrix A has multiplied every column of matrix B. Then multiply row 2 matrix A by each column of matrix B until all the row 2 entries of AB are complete. And so on. (See the Quick Summary for this unit for examples of matrix operations.) Because this is how matrix multiplication is defined, we can only multiply matrices A and B if the number of entries in row 1 matrix A matches the number of entries in column 1 matrix B, which is to say that the number of columns in matrix A must match the number of rows in matrix B. Thus, a matrix with dimension 3 by 4 (which means 3 rows and 4 columns) can multiply any matrix with dimension 4 by Z (where Z can be any number of columns). The resulting AB would have dimension 3 by Z. This definition also has repercussions on the existence of inverse matrices. Because the multiplications A(A-1) and (A-1)A must result in the same identity matrix, only square matrices have inverses. (If A were a 3 by 4 matrix then A-1 would have to be a 4 by 3 matrix to give a 3 by 3 identity for the result. But if we reverse the order, then (A-1)A would require us to multiply a 4 by 3 matrix with a 3 by 4 matrix, for a 4 by 4 result, NOT the same identity in both cases. A cannot be a non-square matrix if the inverse exists.)
• Describe how to perform the operation using a calculator. Different calculators have different steps, but all require the user to enter the cell values into a matrix with a particular name and store that. You will have to specify the dimensions so that there are enough spaces to be filled. Repeat with another matrix with a different name. Then on the home screen call up the names of the matrices with the operation required, paying particular attention to multiplication since order matters.
• Give at least one example showing how the operation can be used to help you analyze some situation. Students found the difference in auto production of three car companies each quarter for two years and the total production by subtracting or adding matrices. They multiplied the number of houses roofed by three crews in two time periods, by the time required per roof to do two different processes, by these same three crews, to find the total time required by all crews to complete the two different processes.

If you would like to see specific problems from Course 2, Unit 1, a link is provided to Examples of Tasks from Course 2, Unit 1. If you are interested in following up on the Algebra or Discrete Mathematics strands in general, then the Scope and Sequence will help you see where different concepts are introduced. On the Algebra and Discrete Mathematics pages, you can read an explanation of the main algebra and discrete mathematics concepts as they are developed in all four courses.