Course 2, Unit 2 - Matrix Methods
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 Discrete Mathematical Modeling 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
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.) (See pages 83-85.) 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:
(See pages 103-112.)
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 technology.
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, = .