4-3 Multiplying Matrices
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Transcript 4-3 Multiplying Matrices
Objectives:
1. Multiply matrices.
2. 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.