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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

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 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)







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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