The Basics of Matrices
Matrix Addition
Scalar Multiplication
MATH 105: Finite Mathematics
2-4: Matrix Algebra
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Conclusion
The Basics of Matrices
Matrix Addition
Outline
1
The Basics of Matrices
2
Matrix Addition
3
Scalar Multiplication
4
Conclusion
Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Outline
1
The Basics of Matrices
2
Matrix Addition
3
Scalar Multiplication
4
Conclusion
Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Exploring a Matrix
Since we’ve seen that matrices can be useful in solving equations,
it makes sense to become more familiar with them.
Matrix Vocabular
The matrix shown below has 2 rows and 3 columns. Its dimension
is 2 × 3. Any element of the matrix can be located by specifying
the row and column number in which is appears.
a11 a12 a13
a21 a22 a23
Locating Elements
Identify the element in each location.
1
The element in the 1st row, 2nd column
2
The element in the 2nd row, 3rd column
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Exploring a Matrix
Since we’ve seen that matrices can be useful in solving equations,
it makes sense to become more familiar with them.
Matrix Vocabular
The matrix shown below has 2 rows and 3 columns. Its dimension
is 2 × 3. Any element of the matrix can be located by specifying
the row and column number in which is appears.
a11 a12 a13
a21 a22 a23
Locating Elements
Identify the element in each location.
1
The element in the 1st row, 2nd column
2
The element in the 2nd row, 3rd column
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Exploring a Matrix
Since we’ve seen that matrices can be useful in solving equations,
it makes sense to become more familiar with them.
Matrix Vocabular
The matrix shown below has 2 rows and 3 columns. Its dimension
is 2 × 3. Any element of the matrix can be located by specifying
the row and column number in which is appears.
a11 a12 a13
a21 a22 a23
Locating Elements
Identify the element in each location.
1
The element in the 1st row, 2nd column
2
The element in the 2nd row, 3rd column
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Exploring a Matrix
Since we’ve seen that matrices can be useful in solving equations,
it makes sense to become more familiar with them.
Matrix Vocabular
The matrix shown below has 2 rows and 3 columns. Its dimension
is 2 × 3. Any element of the matrix can be located by specifying
the row and column number in which is appears.
a11 a12 a13
a21 a22 a23
Locating Elements
Identify the element in each location.
1
The element in the 1st row, 2nd column
2
The element in the 2nd row, 3rd column
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Exploring a Matrix
Since we’ve seen that matrices can be useful in solving equations,
it makes sense to become more familiar with them.
Matrix Vocabular
The matrix shown below has 2 rows and 3 columns. Its dimension
is 2 × 3. Any element of the matrix can be located by specifying
the row and column number in which is appears.
a11 a12 a13
a21 a22 a23
Locating Elements
Identify the element in each location.
1
The element in the 1st row, 2nd column (a12 )
2
The element in the 2nd row, 3rd column
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Exploring a Matrix
Since we’ve seen that matrices can be useful in solving equations,
it makes sense to become more familiar with them.
Matrix Vocabular
The matrix shown below has 2 rows and 3 columns. Its dimension
is 2 × 3. Any element of the matrix can be located by specifying
the row and column number in which is appears.
a11 a12 a13
a21 a22 a23
Locating Elements
Identify the element in each location.
1
The element in the 1st row, 2nd column (a12 )
2
The element in the 2nd row, 3rd column
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Exploring a Matrix
Since we’ve seen that matrices can be useful in solving equations,
it makes sense to become more familiar with them.
Matrix Vocabular
The matrix shown below has 2 rows and 3 columns. Its dimension
is 2 × 3. Any element of the matrix can be located by specifying
the row and column number in which is appears.
a11 a12 a13
a21 a22 a23
Locating Elements
Identify the element in each location.
1
The element in the 1st row, 2nd column (a12 )
2
The element in the 2nd row, 3rd column (a23 )
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Another Example
Example
Use the matrix A below to answer

1
A= 2
−1
the following questions.

7 5
4 3
4 0
1
What is the dimension of this matrix?
2
What is the entry in the 1st row, 2nd column?
3
What is the entry in the 3rd row, 1st column?
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Another Example
Example
Use the matrix A below to answer

1
A= 2
−1
the following questions.

7 5
4 3
4 0
1
What is the dimension of this matrix?
2
What is the entry in the 1st row, 2nd column?
3
What is the entry in the 3rd row, 1st column?
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Another Example
Example
Use the matrix A below to answer

1
A= 2
−1
the following questions.

7 5
4 3
4 0
1
What is the dimension of this matrix? (3 × 3)
2
What is the entry in the 1st row, 2nd column?
3
What is the entry in the 3rd row, 1st column?
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Another Example
Example
Use the matrix A below to answer

1
A= 2
−1
the following questions.

7 5
4 3
4 0
1
What is the dimension of this matrix? (3 × 3)
2
What is the entry in the 1st row, 2nd column?
3
What is the entry in the 3rd row, 1st column?
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Another Example
Example
Use the matrix A below to answer

1
A= 2
−1
the following questions.

7 5
4 3
4 0
1
What is the dimension of this matrix? (3 × 3)
2
What is the entry in the 1st row, 2nd column? (7)
3
What is the entry in the 3rd row, 1st column?
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Another Example
Example
Use the matrix A below to answer

1
A= 2
−1
the following questions.

7 5
4 3
4 0
1
What is the dimension of this matrix? (3 × 3)
2
What is the entry in the 1st row, 2nd column? (7)
3
What is the entry in the 3rd row, 1st column?
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Another Example
Example
Use the matrix A below to answer

1
A= 2
−1
the following questions.

7 5
4 3
4 0
1
What is the dimension of this matrix? (3 × 3)
2
What is the entry in the 1st row, 2nd column? (7)
3
What is the entry in the 3rd row, 1st column? (−1)
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Special Types of Matrices
Two types of matrix are of particular interest.
Row Vector
A row vector is a 1 × n matrix where n is any integer greater than
zero.
Column Vector
A column vector is an n × 1 matrix where n is any integer greater
than zero.
An Example Row Vector
The matrix below is a 1 × 4
row vector.
U = 1 2 4 −3
An Example Column Vector
The matrix below is a 2 × 1
column vector.
2
V =
3
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Special Types of Matrices
Two types of matrix are of particular interest.
Row Vector
A row vector is a 1 × n matrix where n is any integer greater than
zero.
Column Vector
A column vector is an n × 1 matrix where n is any integer greater
than zero.
An Example Row Vector
The matrix below is a 1 × 4
row vector.
U = 1 2 4 −3
An Example Column Vector
The matrix below is a 2 × 1
column vector.
2
V =
3
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Special Types of Matrices
Two types of matrix are of particular interest.
Row Vector
A row vector is a 1 × n matrix where n is any integer greater than
zero.
Column Vector
A column vector is an n × 1 matrix where n is any integer greater
than zero.
An Example Row Vector
The matrix below is a 1 × 4
row vector.
U = 1 2 4 −3
An Example Column Vector
The matrix below is a 2 × 1
column vector.
2
V =
3
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Special Types of Matrices
Two types of matrix are of particular interest.
Row Vector
A row vector is a 1 × n matrix where n is any integer greater than
zero.
Column Vector
A column vector is an n × 1 matrix where n is any integer greater
than zero.
An Example Row Vector
The matrix below is a 1 × 4
row vector.
U = 1 2 4 −3
An Example Column Vector
The matrix below is a 2 × 1
column vector.
2
V =
3
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Special Types of Matrices
Two types of matrix are of particular interest.
Row Vector
A row vector is a 1 × n matrix where n is any integer greater than
zero.
Column Vector
A column vector is an n × 1 matrix where n is any integer greater
than zero.
An Example Row Vector
The matrix below is a 1 × 4
row vector.
U = 1 2 4 −3
An Example Column Vector
The matrix below is a 2 × 1
column vector.
2
V =
3
The Basics of Matrices
Matrix Addition
Outline
1
The Basics of Matrices
2
Matrix Addition
3
Scalar Multiplication
4
Conclusion
Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Adding Matrices
When mathematicians introduce a new object, one of the first
things they like to do with them is figure out how to combine them.
Adding Matrices
The sum of two matrices, A + B, of the same dimension is the
matrix consisting of the sum of corresponding entries from A and
B. The resulting matrix has the same dimension as both original
matrices A and B
Things to Notice:
1
You can only add matrices of the same dimension.
2
The new matrix will have this same dimension.
3
Add matrices element-by-element.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Adding Matrices
When mathematicians introduce a new object, one of the first
things they like to do with them is figure out how to combine them.
Adding Matrices
The sum of two matrices, A + B, of the same dimension is the
matrix consisting of the sum of corresponding entries from A and
B. The resulting matrix has the same dimension as both original
matrices A and B
Things to Notice:
1
You can only add matrices of the same dimension.
2
The new matrix will have this same dimension.
3
Add matrices element-by-element.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Adding Matrices
When mathematicians introduce a new object, one of the first
things they like to do with them is figure out how to combine them.
Adding Matrices
The sum of two matrices, A + B, of the same dimension is the
matrix consisting of the sum of corresponding entries from A and
B. The resulting matrix has the same dimension as both original
matrices A and B
Things to Notice:
1
You can only add matrices of the same dimension.
2
The new matrix will have this same dimension.
3
Add matrices element-by-element.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Adding Matrices
When mathematicians introduce a new object, one of the first
things they like to do with them is figure out how to combine them.
Adding Matrices
The sum of two matrices, A + B, of the same dimension is the
matrix consisting of the sum of corresponding entries from A and
B. The resulting matrix has the same dimension as both original
matrices A and B
Things to Notice:
1
You can only add matrices of the same dimension.
2
The new matrix will have this same dimension.
3
Add matrices element-by-element.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Adding Matrices
When mathematicians introduce a new object, one of the first
things they like to do with them is figure out how to combine them.
Adding Matrices
The sum of two matrices, A + B, of the same dimension is the
matrix consisting of the sum of corresponding entries from A and
B. The resulting matrix has the same dimension as both original
matrices A and B
Things to Notice:
1
You can only add matrices of the same dimension.
2
The new matrix will have this same dimension.
3
Add matrices element-by-element.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Adding Matrices
When mathematicians introduce a new object, one of the first
things they like to do with them is figure out how to combine them.
Adding Matrices
The sum of two matrices, A + B, of the same dimension is the
matrix consisting of the sum of corresponding entries from A and
B. The resulting matrix has the same dimension as both original
matrices A and B
Things to Notice:
1
You can only add matrices of the same dimension.
2
The new matrix will have this same dimension.
3
Add matrices element-by-element.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Matrix Addition
Matrix Addition
Add each pair of matrices, if possible.
1
2 −1 1
−1 4 1
+
4 3 5
0 −2 3
2
1 2
1
+
7 2
−3
3
4 1
−2 1
+
3 2
1 0
4
−2 1
4 1
+
1 0
3 2
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Matrix Addition
Matrix Addition
Add each pair of matrices, if possible.
1
2 −1 1
−1 4 1
+
4 3 5
0 −2 3
2
1 2
1
+
7 2
−3
3
4 1
−2 1
+
3 2
1 0
4
−2 1
4 1
+
1 0
3 2
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Matrix Addition
Matrix Addition
Add each pair of matrices, if possible.
1
2 −1 1
−1 4 1
+
4 3 5
0 −2 3
2
1 2
1
+
7 2
−3
3
4 1
−2 1
+
3 2
1 0
4
−2 1
4 1
+
1 0
3 2
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Matrix Addition
Matrix Addition
Add each pair of matrices, if possible.
1
2 −1 1
−1 4 1
+
4 3 5
0 −2 3
2
1 2
1
+
7 2
−3
3
4 1
−2 1
+
3 2
1 0
4
−2 1
4 1
+
1 0
3 2
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Matrix Addition
In the last two examples, we got the same answer.
Properties of Matrix Addition
Let A, B, and C be matrices of the same dimension. Then,
1
A + B = B + A (Commutative Property of Addition)
2
A + (B + C ) = (A + B) + C (Associative Property of Addition)
3
A + (−A) = 0 (Inverse Property of Addition)
The Zero Matrix
A matrix of any dimension consisting of all zeros is called a zero
matrix and written 0.
Matrix Subtraction
The difference between two matrices, A − B, is the sum of A and
−B, the matrix obtained by multiplying every entry of B by −1.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Matrix Addition
In the last two examples, we got the same answer.
Properties of Matrix Addition
Let A, B, and C be matrices of the same dimension. Then,
1
A + B = B + A (Commutative Property of Addition)
2
A + (B + C ) = (A + B) + C (Associative Property of Addition)
3
A + (−A) = 0 (Inverse Property of Addition)
The Zero Matrix
A matrix of any dimension consisting of all zeros is called a zero
matrix and written 0.
Matrix Subtraction
The difference between two matrices, A − B, is the sum of A and
−B, the matrix obtained by multiplying every entry of B by −1.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Matrix Addition
In the last two examples, we got the same answer.
Properties of Matrix Addition
Let A, B, and C be matrices of the same dimension. Then,
1
A + B = B + A (Commutative Property of Addition)
2
A + (B + C ) = (A + B) + C (Associative Property of Addition)
3
A + (−A) = 0 (Inverse Property of Addition)
The Zero Matrix
A matrix of any dimension consisting of all zeros is called a zero
matrix and written 0.
Matrix Subtraction
The difference between two matrices, A − B, is the sum of A and
−B, the matrix obtained by multiplying every entry of B by −1.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Matrix Addition
In the last two examples, we got the same answer.
Properties of Matrix Addition
Let A, B, and C be matrices of the same dimension. Then,
1
A + B = B + A (Commutative Property of Addition)
2
A + (B + C ) = (A + B) + C (Associative Property of Addition)
3
A + (−A) = 0 (Inverse Property of Addition)
The Zero Matrix
A matrix of any dimension consisting of all zeros is called a zero
matrix and written 0.
Matrix Subtraction
The difference between two matrices, A − B, is the sum of A and
−B, the matrix obtained by multiplying every entry of B by −1.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Matrix Addition
In the last two examples, we got the same answer.
Properties of Matrix Addition
Let A, B, and C be matrices of the same dimension. Then,
1
A + B = B + A (Commutative Property of Addition)
2
A + (B + C ) = (A + B) + C (Associative Property of Addition)
3
A + (−A) = 0 (Inverse Property of Addition)
The Zero Matrix
A matrix of any dimension consisting of all zeros is called a zero
matrix and written 0.
Matrix Subtraction
The difference between two matrices, A − B, is the sum of A and
−B, the matrix obtained by multiplying every entry of B by −1.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Matrix Addition
In the last two examples, we got the same answer.
Properties of Matrix Addition
Let A, B, and C be matrices of the same dimension. Then,
1
A + B = B + A (Commutative Property of Addition)
2
A + (B + C ) = (A + B) + C (Associative Property of Addition)
3
A + (−A) = 0 (Inverse Property of Addition)
The Zero Matrix
A matrix of any dimension consisting of all zeros is called a zero
matrix and written 0.
Matrix Subtraction
The difference between two matrices, A − B, is the sum of A and
−B, the matrix obtained by multiplying every entry of B by −1.
The Basics of Matrices
Matrix Addition
Outline
1
The Basics of Matrices
2
Matrix Addition
3
Scalar Multiplication
4
Conclusion
Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Multiplication
Now that we’ve introduced addition and subtraction, we turn to
multiplication. There are two types of multiplication which we will
consider in this class. In this section, we look at the first.
Scalar Multiplication
Let A be an n × m matrix and c a real number, called a scalar.
The produce of the matrix A with the scalar c is the m × n matrix
cA whose entries are the produce of c with the corresponding
entries in A.
Things to Notice:
1
The dimension of the new matrix is the same as that of A.
2
This is called Scalar Multiplication because we multiply by the
scalar c, and not by another matrix.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Multiplication
Now that we’ve introduced addition and subtraction, we turn to
multiplication. There are two types of multiplication which we will
consider in this class. In this section, we look at the first.
Scalar Multiplication
Let A be an n × m matrix and c a real number, called a scalar.
The produce of the matrix A with the scalar c is the m × n matrix
cA whose entries are the produce of c with the corresponding
entries in A.
Things to Notice:
1
The dimension of the new matrix is the same as that of A.
2
This is called Scalar Multiplication because we multiply by the
scalar c, and not by another matrix.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Multiplication
Now that we’ve introduced addition and subtraction, we turn to
multiplication. There are two types of multiplication which we will
consider in this class. In this section, we look at the first.
Scalar Multiplication
Let A be an n × m matrix and c a real number, called a scalar.
The produce of the matrix A with the scalar c is the m × n matrix
cA whose entries are the produce of c with the corresponding
entries in A.
Things to Notice:
1
The dimension of the new matrix is the same as that of A.
2
This is called Scalar Multiplication because we multiply by the
scalar c, and not by another matrix.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Multiplication
Now that we’ve introduced addition and subtraction, we turn to
multiplication. There are two types of multiplication which we will
consider in this class. In this section, we look at the first.
Scalar Multiplication
Let A be an n × m matrix and c a real number, called a scalar.
The produce of the matrix A with the scalar c is the m × n matrix
cA whose entries are the produce of c with the corresponding
entries in A.
Things to Notice:
1
The dimension of the new matrix is the same as that of A.
2
This is called Scalar Multiplication because we multiply by the
scalar c, and not by another matrix.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Multiplication
Now that we’ve introduced addition and subtraction, we turn to
multiplication. There are two types of multiplication which we will
consider in this class. In this section, we look at the first.
Scalar Multiplication
Let A be an n × m matrix and c a real number, called a scalar.
The produce of the matrix A with the scalar c is the m × n matrix
cA whose entries are the produce of c with the corresponding
entries in A.
Things to Notice:
1
The dimension of the new matrix is the same as that of A.
2
This is called Scalar Multiplication because we multiply by the
scalar c, and not by another matrix.
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Scalar Multiplication
Examples
Find each scalar produce using the matrices A and B below.
1 3
2 −1
A=
B=
5 −1
7 −4
1
3A
2
−2B
3
2A − 3B
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Scalar Multiplication
Examples
Find each scalar produce using the matrices A and B below.
1 3
2 −1
A=
B=
5 −1
7 −4
1
3A
2
−2B
3
2A − 3B
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Scalar Multiplication
Examples
Find each scalar produce using the matrices A and B below.
1 3
2 −1
A=
B=
5 −1
7 −4
1
3A
2
−2B
3
2A − 3B
3
9
15 −3
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Scalar Multiplication
Examples
Find each scalar produce using the matrices A and B below.
1 3
2 −1
A=
B=
5 −1
7 −4
1
3A
2
−2B
3
2A − 3B
3
9
15 −3
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Scalar Multiplication
Examples
Find each scalar produce using the matrices A and B below.
1 3
2 −1
A=
B=
5 −1
7 −4
1
2
3A
3
9
15 −3
−2B
−4 2
−14 8
3
2A − 3B
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Scalar Multiplication
Examples
Find each scalar produce using the matrices A and B below.
1 3
2 −1
A=
B=
5 −1
7 −4
1
2
3A
3
9
15 −3
−2B
−4 2
−14 8
3
2A − 3B
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Examples of Scalar Multiplication
Examples
Find each scalar produce using the matrices A and B below.
1 3
2 −1
A=
B=
5 −1
7 −4
1
2
3A
3
9
15 −3
−2B
−4 2
−14 8
3
2A − 3B
−1 11
1
5
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Scalar Multiplication
Just as with addition, there are certain properties of scalar
multiplication of which we should be aware.
Properties of Scalar Multiplication
Let k and h be real numbers, A and B matrices of the same
dimension. Then,
1
k(hA) = (hk)A (Associative Property of Scalar Multiplication)
2
(k + h)A = kA + hA (Distributive Property I)
3
k(A + B) = kA + kB (Distributive Property II)
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Scalar Multiplication
Just as with addition, there are certain properties of scalar
multiplication of which we should be aware.
Properties of Scalar Multiplication
Let k and h be real numbers, A and B matrices of the same
dimension. Then,
1
k(hA) = (hk)A (Associative Property of Scalar Multiplication)
2
(k + h)A = kA + hA (Distributive Property I)
3
k(A + B) = kA + kB (Distributive Property II)
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Scalar Multiplication
Just as with addition, there are certain properties of scalar
multiplication of which we should be aware.
Properties of Scalar Multiplication
Let k and h be real numbers, A and B matrices of the same
dimension. Then,
1
k(hA) = (hk)A (Associative Property of Scalar Multiplication)
2
(k + h)A = kA + hA (Distributive Property I)
3
k(A + B) = kA + kB (Distributive Property II)
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Scalar Multiplication
Just as with addition, there are certain properties of scalar
multiplication of which we should be aware.
Properties of Scalar Multiplication
Let k and h be real numbers, A and B matrices of the same
dimension. Then,
1
k(hA) = (hk)A (Associative Property of Scalar Multiplication)
2
(k + h)A = kA + hA (Distributive Property I)
3
k(A + B) = kA + kB (Distributive Property II)
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Properties of Scalar Multiplication
Just as with addition, there are certain properties of scalar
multiplication of which we should be aware.
Properties of Scalar Multiplication
Let k and h be real numbers, A and B matrices of the same
dimension. Then,
1
k(hA) = (hk)A (Associative Property of Scalar Multiplication)
2
(k + h)A = kA + hA (Distributive Property I)
3
k(A + B) = kA + kB (Distributive Property II)
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Verifying Properties
Example
Verify these properties using the scalars k = 2, h = 3 and the
matrices A and B shown below.
0 2 3
−1 2 5
A=
B=
1 −1 5
0 1 0
1
2(3A) = 6A
2
(2 + 3)B = 2B + 3B
3
2(A + B) = 2A + 2B
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Verifying Properties
Example
Verify these properties using the scalars k = 2, h = 3 and the
matrices A and B shown below.
0 2 3
−1 2 5
A=
B=
1 −1 5
0 1 0
1
2(3A) = 6A
2
(2 + 3)B = 2B + 3B
3
2(A + B) = 2A + 2B
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Verifying Properties
Example
Verify these properties using the scalars k = 2, h = 3 and the
matrices A and B shown below.
0 2 3
−1 2 5
A=
B=
1 −1 5
0 1 0
1
2(3A) = 6A
2
(2 + 3)B = 2B + 3B
3
2(A + B) = 2A + 2B
Conclusion
The Basics of Matrices
Matrix Addition
Outline
1
The Basics of Matrices
2
Matrix Addition
3
Scalar Multiplication
4
Conclusion
Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Important Concepts
Things to Remember from Section 2-4
1
Matrix Dimensions and Element Locations
2
Rules for Matrix Addition
3
Rules for Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Important Concepts
Things to Remember from Section 2-4
1
Matrix Dimensions and Element Locations
2
Rules for Matrix Addition
3
Rules for Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Important Concepts
Things to Remember from Section 2-4
1
Matrix Dimensions and Element Locations
2
Rules for Matrix Addition
3
Rules for Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Important Concepts
Things to Remember from Section 2-4
1
Matrix Dimensions and Element Locations
2
Rules for Matrix Addition
3
Rules for Scalar Multiplication
Conclusion
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Next Time. . .
In section 2-5 we will learn how to multiply two matrices together.
This will give us the ability to solve systems of equations using
matrices in a new way.
For next time
Read section 2-5
The Basics of Matrices
Matrix Addition
Scalar Multiplication
Conclusion
Next Time. . .
In section 2-5 we will learn how to multiply two matrices together.
This will give us the ability to solve systems of equations using
matrices in a new way.
For next time
Read section 2-5