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.