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The Humongous Book of Algebra Problems by W. Michael Kelley

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Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
188
Multiplying Matrices
Not as easy as adding or subtracting them
9.16 If matrix A has order c × d, matrix B has order m × n, and the product matrix
P = A B exists, what condition must be met by the dimensions of A and B?
What is the order of P?
If the product A B exists, then the number of columns in matrix A must equal
the number of rows in matrix B. In this example, d = m.
The dimensions of the product matrix are dictated by the dimensions of A and
B as well. Matrix P will have the same number of rows as matrix A and the same
number of columns as matrix B. Therefore, matrix P is c × n.
Note: Problems 9.17–9.19 refer to matrices A and B (as defined below) and matrix P, such
that P = A B.
9.17 Calculate element p
11
of matrix P.
To calculate an element p
ij
of the product matrix P, multiply each element in the
ith row of A by the corresponding element in the jth column of B and add the
products together.
Note: Problems 9.17–9.19 refer to matrices A and B (as defined below) and matrix P, such
that P = A B.
9.18 Calculate element p
12
of matrix P.
Multiply each element in the first row of A by the corresponding element in the
second column of B.
If there arent
as many rows in the
second matrix as there
are columns in the
rst matrix, you cant
multiply the matrices.
You are
calculating p
11
,
so you’ll multiply
the numbers in the
rst row of A (starting
at the left side) by the
numbers in the rst
column of B (starting
at the top), one at
a time.
Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
189
Note: Problems 9.17–9.19 refer to matrices A and B (as defined below) and matrix P, such
that P = A B.
9.19 Calculate elements p
21
and p
22
to complete the product matrix P.
To calculate p
21
, multiply each element in the second row of A by the
corresponding element in the first column of B.
To calculate p
22
, multiply each element in the second row of A by the
corresponding element in the second column of B.
Create matrix P, placing elements p
11
, p
12
, p
21
, and p
22
in the positions indicated.
9.20 Multiply the matrices: .
Calculate each element p
ij
of the product matrix P by multiplying each element
in the ith row of the left matrix by the corresponding element in the jth column
of the right matrix and then summing those products, as explained in Problems
9.17–9.19.
Remember,
the little
numbers tell you
what row and
column an element
is in, so p
21
is in row 2
column 1 and p
22
is
in row 2 column 2.
Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
190
Note: Problems 9.21–9.22 refer to matrices A and B (as defined below), and matrix P, such
that P = A B.
9.21 Explain why P exists and calculate element p
42
.
The product P = A B exists because the number of columns in matrix A is
equal to the number of rows in matrix B.
To calculate element p
42
of matrix P, multiply each element in the fourth row of
A by the corresponding element in the second column of B and add the results.
Note: Problems 9.21–9.22 refer to matrices A and B (as defined below), and matrix P, such
that P = A B.
9.22 Generate matrix P.
The product matrix has the same number of rows as matrix A and the same
number of columns as matrix B. Therefore, P has order 4 × 3. Calculate each
element of P using the technique described in Problem 9.21.
A has two columns
and B has two rows.
You can mul-
tiply A B, but
you cant multiply
B A, because the
number of columns in B
(3) does not equal the
number of rows in
A (4).

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