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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

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.

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


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 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 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 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 Products

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

  • Example: A3x2 and B3x2


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 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 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 Matrices

  • Find RS if


Multiplying Matrices

  • Find RS if


Multiplying Matrices

  • Find RS if

    (first row, first column)


Multiplying Matrices

  • Find RS if

    (first row, second column)


Multiplying Matrices

  • Find RS if

    (second row, first column)


Multiplying Matrices

  • Find RS if

    (second row, second column)


Multiplying Matrices

  • Find UV if


Multiplying Matrices

  • Find UV if


Multiplying Matrices

  • Find UV if


Multiplying Matrices

  • Find UV if


Multiplying Matrices

  • Find UV if


Properties of Multiplying Matrices

  • Matrix multiplication is NOT commutative.

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


AB≠BA in Matrices

  • Find KL if


AB≠BA in Matrices

  • Find KL if


AB≠BA in Matrices

  • Find KL if


AB≠BA in Matrices

  • Find KL if


AB≠BA in Matrices

  • Find KL if


AB≠BA in Matrices

  • Find LK if


AB≠BA in Matrices

  • Find LK if


AB≠BA in Matrices

  • Find LK if


AB≠BA in Matrices

  • Find LK if


AB≠BA in Matrices

  • As you can see, multiplication is NOT commutative.

  • The order of multiplication matters.


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

  • Find A(B+C) if


Distributive Property

  • Find A(B+C) if


Distributive Property

  • Find A(B+C) if


Distributive Property

  • Find A(B+C) if


Distributive Property

  • Find A(B+C) if


Distributive Property

  • Find A(B+C) if


Distributive Property

  • Find A(B+C) if


Distributive Property

  • Find AB+AC if


Distributive Property

  • Find AB+AC if


Distributive Property

  • Find AB+AC if


Distributive Property

  • Find AB+AC if


Distributive Property

  • Find AB+AC if


Distributive Property

  • Find AB+AC if


Distributive Property

  • Find AB+AC if


Distributive Property

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


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