Matrix Multiplication

If two matrices $A(m\times n),B(n\times p)$ are multiplied with each other

$$\underset{A}{\underbrace{\left[\begin{array}{ccc}& & \\ & & \\ & & \end{array}\right]}}\underset{B}{\underbrace{\left[\begin{array}{ccc}& & \\ & & \end{array}\right]}}=====================\underset{C(m\times p)}{\underbrace{\left[\begin{array}{ccc}& & \\ & & \\ & & \end{array}\right]}}$$

The point $a_{i,k}$ can be calculated as

$$a\_i,j=\text{row}i\_A\cdot \text{col}j\_B=\sum \_{i=n}^{k}a\_i,nb\_n,j$$

The columns of C are combinations of the columns of $A$ whereas the rows of $C$ are combinations of the rows of $B$.

Column and Row Multiplication

Multiplying Columns by Rows gives an entire matrix. The results are each multiples of the rows of the initial matrix’ values.

\begin{bmatrix} 2 \ 3 \ 4 \\ \end{bmatrix} \begin{bmatrix} 1 & 6 \end{bmatrix} \= \begin{bmatrix} 2 & 12 \\ 3 & 18 \\ 4 & 24 \end{bmatrix}

Row Multiplication

Row times a Matrix results in a row

$$\underbrace{\left[\begin{array}{ccc}-& -& -\\ -& -& -\\ -& -& -\end{array}\right]}\_\text{Rows 1 through 3}\left[\begin{array}{ccc}1& 2& 7\end{array}\right]=============\left[\begin{array}{ccc}1\times \text{Row 1},& 2\times \text{Row 2},& 7\times \text{Row 3}\end{array}\right]$$

Column Multiplication

Column times a Matrix results in a column

$$\underbrace{\left[\begin{array}{ccc}-& -& -\\ -& -& -\\ -& -& -\end{array}\right]}\_\text{Columns 1 through 3}\left[\begin{array}{c}1\\ 2\\ 7\end{array}\right]=============\left[\begin{array}{c}1\times \text{Column 1}\\ 2\times \text{Column 2}\\ 7\times \text{Column 3}\end{array}\right]$$