2. Chapter 7: Matrices and Systems of Equations and Inequalities
7.2 Solution of Linear Systems in Three Variables
7.3 Solution of Linear Systems by Row Transformations
7.4 Matrix Properties and Operations
7.5 Determinants and Cramer’s Rule
7.6 Solution of Linear Systems by Matrix Inverses
7.7 Systems of Inequalities and Linear Programming
7.8 Partial Fractions

3. 7.4 Matrix Properties and Operations
Matrices are classified by their dimensions:
the number of rows by the number of columns.
A matrix with m rows and n columns has dimension m × n.
e.g. The matrix has dimension 2×3.
A square matrix has the same number of rows as it does columns. The dimension of a square matrix is n × n.

4. 7.4 Classifying Matrices by Dimension
Example Find the dimension of each matrix.
(a) The matrix is a 3 × 2 matrix.
(b) The matrix is a 3 × 3 square matrix.
The matrix is a 1 × 5 row
matrix.

5. 7.4 Determining Equality of Matrices
Example
Solution Two matrices are equal if they have the
same dimension and if corresponding elements,
position by position, are equal. This is true in this
case if 2 = x, 1 = y, p = –1, and q = 0.

6. 7.4 Matrix Addition
Example Find each sum.
The sum of two m × n matrices A and B is the m × n matrix A + B in which each element is the sum of the corresponding elements of A and B.

7. 7.4 Matrix Addition
Analytic Solution
Graphing Calculator Solution

8. 7.4 Matrix Addition Analytic Solution
Graphing Calculator Solution The calculator
returns a dimension mismatch error.

9. 7.4 The Zero Matrix
A matrix with only zero elements is called a zero matrix. For example, [ ] is the 1 × 3 zero matrix while
is the 2 × 3 zero matrix.
The elements of matrix –A are the additive inverses of the elements of matrix A. For example, if

10. 7.4 Matrix Subtraction
Example Find the difference of
Solution
If A and B are matrices with the same dimension, then A – B = A + (– B).

11. 7.4 Matrix Multiplication by a Scalar
If a matrix A is added to itself, each element is twice as large as the corresponding element of A.
In the last expression, the 2 in front of the matrix is called a scalar.
A scalar is a special name for a real number.

12. 7.4 Matrix Multiplication by a Scalar
Example Perform the multiplication
Solution
The product of a scalar k and a matrix A is the matrix kA, each of whose elements is k times the corresponding elements of A.

13. 7.4 Matrix Multiplication
Example Suppose you are the manager of a video store
and receive the following order from two distributors: from
Wholesale Enterprises, 2 videotapes, 7 DVDs, and 5 video
games; from Discount Distributors, 4 videotapes, 6 DVDs,
and 9 video games. We can organize the information in table
format and convert it to a matrix. or

14. 7.4 Matrix Multiplication
Suppose each videotape costs the store $12, each DVD costs
$18, and each video game costs $9. To find the total cost of
the products from Wholesale Enterprises, we multiply as
follows.
The products from Wholesale Enterprises cost a total of $195.

15. 7.4 Matrix Multiplication
The result is the sum of three products:
2($12) + 7($18) + 5($9) = $195.
In the same way, using the second row of the matrix and the three costs gives the total from Discount Distributors:
4($12) + 6($18) + 9($9) = $237.
The total costs from the distributors can be written as a
column matrix The product of matrices can be written as

16. 7.4 Matrix Multiplication
The product AB can be found only if the number of columns of A is the same as the number of rows of B.
The product AB of an m × n matrix A and an n × k matrix B is
found as follows:
To get the ith row, jth column element of AB, multiply each element in the ith row of A by the corresponding element in the jth column of B. The sum of these products will give the element of row i, column j of AB. The dimension of AB is m × k.

17. 7.4 Matrix Multiplication
Example Find the product AB of the two matrices
Analytic Solution A has dimension 2 × 3 and B
has dimension 3 × 2, so they are compatible for
multiplication. The product AB has dimension 2 × 2.

18. 7.4 Matrix Multiplication

19. 7.4 Matrix Multiplication
Example Use the graphing calculator to find the
product BA of the two matrices from the previous
problem.
Graphing Calculator Solution Notice AB  BA.

20. 7.4 Applying Matrix Algebra
Example A contractor builds three kinds of houses, models
X, Y, and Z, with a choice of two styles, colonial or ranch.
Matrix A below shows the number of each kind of house the
contractor is planning to build for a new 100-home
subdivision. The amounts are shown in matrix B, while matrix
C gives the cost in dollars for each kind of material. Concrete
is measured in cubic yards, lumber in 1000 board feet, brick in
1000s, and shingles in 100 square feet.
Colonial
Ranch

21. 7.4 Applying Matrix Algebra
What is the total cost of materials for all houses of each model?
How much of each of the four kinds of material must be ordered?
Use a graphing calculator to find the total cost of the materials.
Concrete
Lumber
Brick
Shingles
Cost per Unit

22. 7.4 Applying Matrix Algebra
Solution
To find the materials cost for each model, first
find AB, the total amount of each material needed for
all the houses of each model.
Concrete
Lumber
Brick
Shingles

23. 7.4 Applying Matrix Algebra
Multiplying the total amount of materials matrix AB
and the cost matrix C gives the total cost of materials.
Cost

24. 7.4 Applying Matrix Algebra
The totals of the columns of matrix AB will give a matrix
whose elements represent the total amounts of each material
needed for the subdivision. Call this matrix D, and write it as a
row matrix.
The total cost of all materials is given by the product
of matrix C, the cost matrix, and matrix D, the total amounts
matrix. The total cost of the materials is $188,400.