Matrix multiplication

Compatibility

• You can only multiply two matrices together if the number of columns in the first matrix is equal to the number of rows in the second matrix.
• If A is an m x n matrix and B is a p x q matrix, then A and B can be multiplied together IF n = p.
• The resulting matrix will have an order of m x q.

Commutitive

• Matrix multiplication is not commutative.
• The order in which we multiply matrices matters:

Associative

• IF multiple matrices can be multiplied together (i.e. their orders are compatible), matrix multiplication is associative.
• We can group matrices in any way when multiplying:

Multiplying matrices

Evaluate where and

• Multiply each element of the first row of A by the corresponding element of the column of B:
• Add them up:
  • The top element of the resulting matrix will be .
• Now, multiply each element of the second row of A by the corresponding element of the column of B:
• Add them up:
  • The bottom element of the resulting matrix will be .
• So,
• Answer:

Evaluate

• Multiply each element of the first row of the first matrix by the corresponding element of the column of the second matrix:
• Add them up:
  • The top-left element of the resulting matrix will be .
• Now, multiply each element of the first row of the first matrix by the corresponding element of the second column of the second matrix:
• Add them up:
  • The top-right element of the resulting matrix will be .
• Next, multiply each element of the second row of the first matrix by the corresponding element of the first column of the second matrix:
• Add them up:
  • The bottom-left element of the resulting matrix will be .
• Finally, multiply each element of the second row of the first matrix by the corresponding element of the second column of the second matrix:
• Add them up:
  • The bottom-right element of the resulting matrix will be .
• So,
• Answer: