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 aren’t

as many rows in the

second matrix as there

are columns in the

rst matrix, you can’t

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 can’t 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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