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

Multiplying Matrices Find RS if

Multiplying Matrices Find RS if

(first row, first column) Multiplying Matrices Find RS if (first row, first column)

(first row, second column) Multiplying Matrices Find RS if (first row, second column)

(second row, first column) Multiplying Matrices Find RS if (second row, first column)

(second row, second 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.