- What are the dimensions of any matrix that translates the vertices of a heptagon on the coordinate plane? Explain
the significance of the numbers you used.
Answer: 7×2 The number of rows is seven because each of the seven vertices of the heptagon must be moved. The
second number is two because each vertex has anx−coordinate and ay−coordinate that must be changed.
- The additive identity for real numbers is the number zero because:
a+ 0 =a, for all real numbersa.
What is the additive identity for 3×2 matrices?
Answer:
0 0
0 0
0 0
- If the matrix
2 − 3
2 − 4
2 − 3
were added to matrix a 3×2 matrix containing the vertices of a triangle would the resultingtransformation be an isometry? Why or why not?
Answer: No, in an isometry each point must be moved the same distance in order to preserve size and shape. This
matrix moves the second vertex farther down than it moves the other two vertices.
Reflections
Matrix Multiplication – It will take some time and practice for students to become proficient with matrix multipli-
cation. On first inspection of the formula it is sometimes hard to see where all the numbers are coming from and
where they are going. For many students a spatial representation is more useful. Here are some guidelines that will
help students master matrix multiplication.
a. Use the rows of the first matrix and the columns of the second matrix.
b. Move across and down using each number only once.
c. The resulting sum of the products goes in the slot determined by the row of the first matrix and the column of
the second matrix.Think Don’t Memorize –Many students will try to learn the reflection matrices using rote memory. This is difficult
to begin with since the matrices are fairly similar, but it is also not a good method of learning the material because
the knowledge will not last. As soon as the students stop regularly using the matrices, they will be forgotten. Instead
have the students think about why the matrices produce their intended effect. When the students really understand
what is happening, they will not need to memorize patterns of ones, negative ones, and zeros. The knowledge will
be long lasting, and they will be able to develop new matrices that represent other types of transformation and other
operations.
Additional Exercises:
- Let matrixA=
[
1 2
2 1
]
and matrixB=[
4 3
3 4
]
. CalculateABandBA. What do you notice about the products?
Will this happen with all 2×2 matrices? What is special about matrixAand matrixBthat allowed this special
result?
Answer:AB=BA=
[
10 11
11 10
]
. MatrixAandBcommute. This property does not hold for most matrices. These
matrices commute because they are symmetric matrices. A symmetric matrix is one in which the rows and columns
of the matrix are the same.
Chapter 2. Geometry TE - Common Errors