Course 2, Unit 2 - Matrix Methods

Overview
In the Matrix Methods unit, students learn to use matrices to organize and display data, to operate on matrices in a variety of ways (including summing rows and columns, comparing two rows, finding the mean of rows and columns, scalar multiplication, and addition, subtraction, and multiplication of two matrices), and to interpret the results in context. These ideas are developed in a variety of contexts, giving students an appreciation of the widespread use of matrices. In addition, this unit is rich in connections to other Discrete Mathematics units, such as Vertex-Edge Graphs in Course 1, and Geometry units such as Coordinate Methods in Course 2.

Key Ideas from Course 2, Unit 2

• Matrix: A rectangular array of rows and columns used to organize information.

• Dimension: A matrix has dimension 2 by 3 if it has 2 rows and 3 columns. A square matrix has the same number of rows as columns.

• Matrix operations: Two matrices are added by adding corresponding cells; thus the entries in row i column j are added to get the (i, j) cell in the resulting matrix. (Likewise for subtraction.) Two matrices are multiplied by multiplying each entry of each row of the first matrix by the corresponding entry of each column in the second matrix. Thus:

  
  

• Inverse: A matrix A has an inverse A^-1 if A(A^-1) = (A^-1)A = I, the identity matrix. For example, if A = , then A^-1 = . Also, A(A^-1) = I = .

• Square matrices: The only matrices to possibly have inverses. Students can find inverses, if they exist, by using a calculator. For a 2 by 2 matrix, students have a formula. If A = , then A^-1 = .

• Matrix equation of the form Ax = B: This equation can be solved by multiplying each side of the equation by the inverse of matrix A, giving x = A^-1(B). This can be used to solve systems of equations in more than one variable. For example, can be rewritten as a matrix equation,  = . Which can be solved by multiplying both sides on the left by the inverse of A,  = . So,  = .