MATH 105: Finite Mathematics 2-4: Matrix Algebra Prof. Jonathan - - PowerPoint PPT Presentation

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

MATH 105: Finite Mathematics 2-4: Matrix Algebra

• Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Outline

1

The Basics of Matrices

2

Matrix Addition

3

Scalar Multiplication

4

Conclusion

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Outline

1

The Basics of Matrices

2

Matrix Addition

3

Scalar Multiplication

4

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 Conclusion

Another Example

Example Use the matrix A below to answer the following questions. A =   1 7 5 2 4 3 −1 4  

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?

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Another Example

Example Use the matrix A below to answer the following questions. A =   1 7 5 2 4 3 −1 4  

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?

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Another Example

Example Use the matrix A below to answer the following questions. A =   1 7 5 2 4 3 −1 4  

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?

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Another Example

Example Use the matrix A below to answer the following questions. A =   1 7 5 2 4 3 −1 4  

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?

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Another Example

Example Use the matrix A below to answer the following questions. A =   1 7 5 2 4 3 −1 4  

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?

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Another Example

Example Use the matrix A below to answer the following questions. A =   1 7 5 2 4 3 −1 4  

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?

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Another Example

Example Use the matrix A below to answer the following questions. A =   1 7 5 2 4 3 −1 4  

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)

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. V = 2 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. V = 2 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. V = 2 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. V = 2 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. V = 2 3

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Outline

1

The Basics of Matrices

2

Matrix Addition

3

Scalar Multiplication

4

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 Conclusion

Examples of Matrix Addition

Matrix Addition Add each pair of matrices, if possible.

1

2 −1 1 4 3 5

• +

−1 4 1 −2 3

• 2

1 2 7 2

• +

1 −3

• 3

4 1 3 2

• +

−2 1 1

• 4

−2 1 1

• +

4 1 3 2

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Matrix Addition

Matrix Addition Add each pair of matrices, if possible.

1

2 −1 1 4 3 5

• +

−1 4 1 −2 3

• 2

1 2 7 2

• +

1 −3

• 3

4 1 3 2

• +

−2 1 1

• 4

−2 1 1

• +

4 1 3 2

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Matrix Addition

Matrix Addition Add each pair of matrices, if possible.

1

2 −1 1 4 3 5

• +

−1 4 1 −2 3

• 2

1 2 7 2

• +

1 −3

• 3

4 1 3 2

• +

−2 1 1

• 4

−2 1 1

• +

4 1 3 2

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Matrix Addition

Matrix Addition Add each pair of matrices, if possible.

1

2 −1 1 4 3 5

• +

−1 4 1 −2 3

• 2

1 2 7 2

• +

1 −3

• 3

4 1 3 2

• +

−2 1 1

• 4

−2 1 1

• +

4 1 3 2

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 Scalar Multiplication Conclusion

Outline

1

The Basics of Matrices

2

Matrix Addition

3

Scalar Multiplication

4

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 Conclusion

Examples of Scalar Multiplication

Examples Find each scalar produce using the matrices A and B below. A = 1 3 5 −1

• B =

2 −1 7 −4

• 1 3A

2 −2B 3 2A − 3B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Scalar Multiplication

Examples Find each scalar produce using the matrices A and B below. A = 1 3 5 −1

• B =

2 −1 7 −4

• 1 3A

2 −2B 3 2A − 3B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Scalar Multiplication

Examples Find each scalar produce using the matrices A and B below. A = 1 3 5 −1

• B =

2 −1 7 −4

• 1 3A

3 9 15 −3

• 2 −2B

3 2A − 3B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Scalar Multiplication

Examples Find each scalar produce using the matrices A and B below. A = 1 3 5 −1

• B =

2 −1 7 −4

• 1 3A

3 9 15 −3

• 2 −2B

3 2A − 3B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Scalar Multiplication

Examples Find each scalar produce using the matrices A and B below. A = 1 3 5 −1

• B =

2 −1 7 −4

• 1 3A

3 9 15 −3

• 2 −2B

−4 2 −14 8

• 3 2A − 3B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Scalar Multiplication

Examples Find each scalar produce using the matrices A and B below. A = 1 3 5 −1

• B =

2 −1 7 −4

• 1 3A

3 9 15 −3

• 2 −2B

−4 2 −14 8

• 3 2A − 3B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Examples of Scalar Multiplication

Examples Find each scalar produce using the matrices A and B below. A = 1 3 5 −1

• B =

2 −1 7 −4

• 1 3A

3 9 15 −3

• 2 −2B

−4 2 −14 8

• 3 2A − 3B

−1 11 1 5

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 Conclusion

Verifying Properties

Example Verify these properties using the scalars k = 2, h = 3 and the matrices A and B shown below. A = 2 3 1 −1 5

• B =

−1 2 5 1

• 1 2(3A) = 6A

2 (2 + 3)B = 2B + 3B 3 2(A + B) = 2A + 2B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Verifying Properties

Example Verify these properties using the scalars k = 2, h = 3 and the matrices A and B shown below. A = 2 3 1 −1 5

• B =

−1 2 5 1

• 1 2(3A) = 6A

2 (2 + 3)B = 2B + 3B 3 2(A + B) = 2A + 2B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Verifying Properties

Example Verify these properties using the scalars k = 2, h = 3 and the matrices A and B shown below. A = 2 3 1 −1 5

• B =

−1 2 5 1

• 1 2(3A) = 6A

2 (2 + 3)B = 2B + 3B 3 2(A + B) = 2A + 2B

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

Outline

1

The Basics of Matrices

2

Matrix Addition

3

Scalar Multiplication

4

Conclusion

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

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

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

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

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

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

The Basics of Matrices Matrix Addition Scalar Multiplication Conclusion

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

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