4 3 multiplying matrices
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4-3 Multiplying Matrices. Objectives: Multiply matrices. Use the properties of matrix multiplication. Multiplying Matrices. You can multiply two matrices if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix.

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4-3 Multiplying Matrices

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4 3 multiplying matrices

4-3 Multiplying Matrices

Objectives:

Multiply matrices.

Use the properties of matrix multiplication.


Multiplying matrices

Multiplying Matrices

  • You can multiply two matrices if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix.

  • When you multiplying two matrices Amxn and Bnxr, the resulting matrix AB is an m x r matrix.


Dimensions of matrix products

Dimensions of Matrix Products

  • Determine whether each matrix product is defined. If so, state the dimensions of the product.

  • Example: A4x6 and B6x2


Dimensions of matrix products1

Dimensions of Matrix Products

  • Determine whether each matrix product is defined. If so, state the dimensions of the product.

  • Example: A4x6 and B6x2

  • Answer: 4x2


Dimensions of matrix products2

Dimensions of Matrix Products

  • Determine whether each matrix product is defined. If so, state the dimensions of the product.

  • Example: A4x6 and B6x2

  • Answer: 4x2

  • Example: A3x4 and B4x2


Dimensions of matrix products3

Dimensions of Matrix Products

  • Determine whether each matrix product is defined. If so, state the dimensions of the product.

  • Example: A4x6 and B6x2

  • Answer: 4x2

  • Example: A3x4 and B4x2

  • Answer: 3x2


Dimensions of matrix products4

Dimensions of Matrix Products

  • Determine whether each matrix product is defined. If so, state the dimensions of the product.

  • Example: A3x2 and B3x2


Dimensions of matrix products5

Dimensions of Matrix Products

  • Determine whether each matrix product is defined. If so, state the dimensions of the product.

  • Example: A3x2 and B3x2

  • Answer: The matrix is not defined.


Dimensions of matrix products6

Dimensions of Matrix Products

  • Determine whether each matrix product is defined. If so, state the dimensions of the product.

  • Example: A3x2 and B3x2

  • Answer: The matrix is not defined.

  • Example: A3x2 and B4x3


Dimensions of matrix products7

Dimensions of Matrix Products

  • Determine whether each matrix product is defined. If so, state the dimensions of the product.

  • Example: A3x2 and B3x2

  • Answer: The matrix is not defined.

  • Example: A3x2 and B4x3

  • Answer: The matrix is not defined.


Multiplying matrices1

Multiplying Matrices

  • Find RS if


Multiplying matrices2

Multiplying Matrices

  • Find RS if


Multiplying matrices3

Multiplying Matrices

  • Find RS if

    (first row, first column)


Multiplying matrices4

Multiplying Matrices

  • Find RS if

    (first row, second column)


Multiplying matrices5

Multiplying Matrices

  • Find RS if

    (second row, first column)


Multiplying matrices6

Multiplying Matrices

  • Find RS if

    (second row, second column)


Multiplying matrices7

Multiplying Matrices

  • Find UV if


Multiplying matrices8

Multiplying Matrices

  • Find UV if


Multiplying matrices9

Multiplying Matrices

  • Find UV if


Multiplying matrices10

Multiplying Matrices

  • Find UV if


Multiplying matrices11

Multiplying Matrices

  • Find UV if


Properties of multiplying matrices

Properties of Multiplying Matrices

  • Matrix multiplication is NOT commutative.

  • This means that if A and B are matrices, AB≠BA.


Ab ba in matrices

AB≠BA in Matrices

  • Find KL if


Ab ba in matrices1

AB≠BA in Matrices

  • Find KL if


Ab ba in matrices2

AB≠BA in Matrices

  • Find KL if


Ab ba in matrices3

AB≠BA in Matrices

  • Find KL if


Ab ba in matrices4

AB≠BA in Matrices

  • Find KL if


Ab ba in matrices5

AB≠BA in Matrices

  • Find LK if


Ab ba in matrices6

AB≠BA in Matrices

  • Find LK if


Ab ba in matrices7

AB≠BA in Matrices

  • Find LK if


Ab ba in matrices8

AB≠BA in Matrices

  • Find LK if


Ab ba in matrices9

AB≠BA in Matrices

  • As you can see, multiplication is NOT commutative.

  • The order of multiplication matters.


Properties of multiplying matrices1

Properties of Multiplying Matrices

Distributive Property

  • If A, B, and C are matrices, then

    • A(B+C)=AB+AC and

    • (B+C)A=BA+CA


Distributive property

Distributive Property

  • Find A(B+C) if


Distributive property1

Distributive Property

  • Find A(B+C) if


Distributive property2

Distributive Property

  • Find A(B+C) if


Distributive property3

Distributive Property

  • Find A(B+C) if


Distributive property4

Distributive Property

  • Find A(B+C) if


Distributive property5

Distributive Property

  • Find A(B+C) if


Distributive property6

Distributive Property

  • Find A(B+C) if


Distributive property7

Distributive Property

  • Find AB+AC if


Distributive property8

Distributive Property

  • Find AB+AC if


Distributive property9

Distributive Property

  • Find AB+AC if


Distributive property10

Distributive Property

  • Find AB+AC if


Distributive property11

Distributive Property

  • Find AB+AC if


Distributive property12

Distributive Property

  • Find AB+AC if


Distributive property13

Distributive Property

  • Find AB+AC if


Distributive property14

Distributive Property

  • As you can see, you can extend the distributive property to matrices.


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