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:
  • \mathbf{A} \times \mathbf{B} \neq \mathbf{B} \times \mathbf{A}

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:
  • (\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = \mathbf{A} \times (\mathbf{B} \times \mathbf{C})

Multiplying matrices

Evaluate \mathbf{A} \times \mathbf{B} where A=\begin{bmatrix} 2 & 5 & 3 \\ -1 & 3 & 2 \end{bmatrix} and B=\begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}

• Multiply each element of the first row of A by the corresponding element of the column of B:
  • 2\times 2 = 4
  • 5\times 1 = 5
  • 3\times -1 = -3
• Add them up:
  • 4 + 5 + -3 = 6
  • The top element of the resulting matrix will be 6.
• Now, multiply each element of the second row of A by the corresponding element of the column of B:
  • -1\times 2 = -2
  • 3\times 1 = 3
  • 2\times -1 = -2
• Add them up:
  • -2 + 3 + -2 = -1
  • The bottom element of the resulting matrix will be -1.
• So, \mathbf{A} \times \mathbf{B} = \begin{bmatrix} 6 \\ -1 \end{bmatrix}
• Answer: \begin{bmatrix} 6 \\ -1 \end{bmatrix}

Evaluate \begin{bmatrix} 5 & 3 \\ 3 & 2 \end{bmatrix}\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}

• Multiply each element of the first row of the first matrix by the corresponding element of the column of the second matrix:
  • 5\times 2 = 10
  • 3\times 1 = 3
• Add them up:
  • 10 + 3 = 13
  • The top-left element of the resulting matrix will be 13.
• Now, multiply each element of the first row of the first matrix by the corresponding element of the second column of the second matrix:
  • 5\times 1 = 5
  • 3\times 3 = 9
• Add them up:
  • 5 + 9 = 14
  • The top-right element of the resulting matrix will be 14.
• Next, multiply each element of the second row of the first matrix by the corresponding element of the first column of the second matrix:
  • 3\times 2 = 6
  • 2\times 1 = 2
• Add them up:
  • 6 + 2 = 8
  • The bottom-left element of the resulting matrix will be 8.
• Finally, multiply each element of the second row of the first matrix by the corresponding element of the second column of the second matrix:
  • 3\times 1 = 3
  • 2\times 3 = 6
• Add them up:
  • 3 + 6 = 9
  • The bottom-right element of the resulting matrix will be 9.
• So, \begin{bmatrix} 5 & 3 \\ 3 & 2 \end{bmatrix}\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} = \begin{bmatrix} 13 & 14 \\ 8 & 9 \end{bmatrix}
• Answer: \begin{bmatrix} 13 & 14 \\ 8 & 9 \end{bmatrix}

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