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 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
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
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, = .