2, Unit 1 - Matrix Models
In the "Matrix Models" 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 "Graph Models"
in Course 1, and Algebra units such as "Linear Models" in Course 1.
Ideas from Course 2, Unit 1
- Matrix: A rectangular array of rows and columns used to organize
- 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
- 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, = .