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4-3 Multiplying MatricesPowerPoint Presentation

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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