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# 6.3 Matrix Operations and Applications

Students will be able to add, subtract, and multiply matrices by a scalar and be able to organize data using matrices.

by

Tweet## Justine Anders

on 7 September 2012#### Transcript of 6.3 Matrix Operations and Applications

6.3 Matrix Operations and Their Applications You will be able to add, subtract, and multiply matrices by a scalar, as well as be able organize data. Notations for Matrices Denoted by a capitalized letter like A, B, or C.

A lowercase letter is inclosed in brackets denotes a matrix.

A=[a ] . i is the ith row and j is the jth column

The order: like RC Cola- Rows then Columns

A matrix of order m x n has m rows and n columns. Matrix Notation A = Example Time What is the order of A?

Identify a and a Two matrices A and B are equal if and only if they have the same order m x n and a =b for i=1,2,....m and j=1,2,.....m Equality of Matrices A =

B=

A= B if and only if x=1, y+1=5, and z=3 Need to have the same order m x n to add and subtract

Matrix addition is very simple; we just add the corresponding elements.

Example 1:

Example 2: Matrix Addition and Subtraction Add or Subtract the Following Matrices Your turn to Practice Multiply each element of the Matrix A by the scalar c. Multiplying by a Scalar Any matrix with all zero entries for the elements is called a zero matrix.

Examples: Properties of Matrix Addition This is called the

Additive Identity Commutative Property of Addition: 3 + 4 = 4 + 3

Associative Property of Addition: (3+4) + 5 = 3 + (4+5)

Additive Identity: 3 + 0 = 0 + 3 = 3

Additive Inverse: 3 + (-3) = (-3) + 3 = 0 Determine if the Commutative Property of Addition, Associative Property of Addition, Additive Identity, and Additive Inverse Properties of Real Numbers hold true for Matrices. If A =

and B=

find -5B and 2A + 3B Example: Associative Property of Addition:

Commutative Property of Addition:

Distributive Property: Review Properties of Real Numbers What would these properties look like for matrices? Associative Property of Scalar Multiplication

(cd)A= c(dA)

Scalar Identity Property

1A = A

Distributive Property

c(A + B) = cA + cB

Distributive Property

(c + d)A = cA + dA Solve for X in the matrix equation 2X + A = B

where A = and B = Solving a Matrix Equation -1 5

0 2 -6 5

9 1 -1 4

3 0 2 -3

5 -6 0 5 3

-2 6 -8 + -2 4 5

7 -9 6 -6 7

2 -3 - -5 6

0 -4 23 12 ij Perform the indicated operations to answer the following questions CLOSURE 5 -2

-3 4

1 6 A= What is the order of A?

Identify a and a 12 31 If A= and B=

find -6B and 3A + 2B -4 1

3 0 -1 -2

8 5 -4 3

7 -6 + 6 -3

2 -4 5 4

-3 7

0 1 -4 8

6 0

-5 3 - 3 2 0

-4 5 -1/5 x y+1

z 6 1 5

3 6 1 2 -5 2

3 6 0 0

0 0 + = -5 2

3 6 REVIEW

Full transcriptA lowercase letter is inclosed in brackets denotes a matrix.

A=[a ] . i is the ith row and j is the jth column

The order: like RC Cola- Rows then Columns

A matrix of order m x n has m rows and n columns. Matrix Notation A = Example Time What is the order of A?

Identify a and a Two matrices A and B are equal if and only if they have the same order m x n and a =b for i=1,2,....m and j=1,2,.....m Equality of Matrices A =

B=

A= B if and only if x=1, y+1=5, and z=3 Need to have the same order m x n to add and subtract

Matrix addition is very simple; we just add the corresponding elements.

Example 1:

Example 2: Matrix Addition and Subtraction Add or Subtract the Following Matrices Your turn to Practice Multiply each element of the Matrix A by the scalar c. Multiplying by a Scalar Any matrix with all zero entries for the elements is called a zero matrix.

Examples: Properties of Matrix Addition This is called the

Additive Identity Commutative Property of Addition: 3 + 4 = 4 + 3

Associative Property of Addition: (3+4) + 5 = 3 + (4+5)

Additive Identity: 3 + 0 = 0 + 3 = 3

Additive Inverse: 3 + (-3) = (-3) + 3 = 0 Determine if the Commutative Property of Addition, Associative Property of Addition, Additive Identity, and Additive Inverse Properties of Real Numbers hold true for Matrices. If A =

and B=

find -5B and 2A + 3B Example: Associative Property of Addition:

Commutative Property of Addition:

Distributive Property: Review Properties of Real Numbers What would these properties look like for matrices? Associative Property of Scalar Multiplication

(cd)A= c(dA)

Scalar Identity Property

1A = A

Distributive Property

c(A + B) = cA + cB

Distributive Property

(c + d)A = cA + dA Solve for X in the matrix equation 2X + A = B

where A = and B = Solving a Matrix Equation -1 5

0 2 -6 5

9 1 -1 4

3 0 2 -3

5 -6 0 5 3

-2 6 -8 + -2 4 5

7 -9 6 -6 7

2 -3 - -5 6

0 -4 23 12 ij Perform the indicated operations to answer the following questions CLOSURE 5 -2

-3 4

1 6 A= What is the order of A?

Identify a and a 12 31 If A= and B=

find -6B and 3A + 2B -4 1

3 0 -1 -2

8 5 -4 3

7 -6 + 6 -3

2 -4 5 4

-3 7

0 1 -4 8

6 0

-5 3 - 3 2 0

-4 5 -1/5 x y+1

z 6 1 5

3 6 1 2 -5 2

3 6 0 0

0 0 + = -5 2

3 6 REVIEW