Teaching Notes 3.23: Identifying Conditions for Multiplying
Two Matrices
Multiplying two matrices can be challenging for many students. They must be able to identify
which matrices can be multiplied and they often have trouble with the concept that
multiplication with matrices is not commutative.- Review the matrix vocabulary, especially row, column, and dimensions. Note that a row is
horizontal and a column is vertical. The number of rows and columns determines the dimen-
sions of a matrix. Thus, a 3×2matrixhasthreerowsandtwocolumns. - Explain the notation:Am×n. This is a type of shorthand for saying MatrixAhasmrows andn
columns. - Review the information and example on the worksheet with your students. Point out that in
the example the productA×Bcan be found. Ask them ifB×Acan be found. Explain that
this is impossible becauseBhas one column andAhas two rows. Emphasize that multiply-
ing matrices is not always commutative. In order to multiply two matrices, the number of
columns in the first matrix must be the same as the number of rows in the second. - Carefully go over the directions for the problems with your students. Be sure that they under-
stand they must find the dimensions of each matrix first before they multiply the matrices.
EXTRA HELP:
Do not assume that multiplication of twomatrices is commutative. The commutative property
applies only if the number of columns in the first matrix is the same as the number of rows in the
second matrix.ANSWER KEY:
(1)P 2 × 2 (2)P 2 × 2 (3)P 2 × 1 (4)Cannot be multiplied (5)Cannot be multiplied (6)P 2 × 1
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(Challenge)Agree. If the number of all of the rows and all of the columns are the same, the
------------------------------------------------------------------------------------------number of columns in the first matrix must be the same as the number of rows in the second.132 THE ALGEBRA TEACHER’S GUIDE