Use adaptive quiz-based learning to study this topic faster and more effectively.

# Multiplication of two matrices

The multiplication $A\times B$ of two matrices $A$ and $B$ is not straightforward. We assume that $A$ has size $m_A\times n_A$ and $B$ has size $m_B\times n_B$.

• Size requirement. We must have $$n_A = m_B$$
• Size of the product The matrix $A\times B$ has size $m_A\times n_B$..
• To compute one entry, we isolate one row in $A$ and one column in $B$. We multiply each pairing terms and we add them. We repeat the process for each row of $A$ and each column of $B$.
$$\begin{pmatrix} 1&0\\ 1&2\\ 0&1 \end{pmatrix} \times \begin{pmatrix} 0&1\\ 0&1\\ 1&0 \end{pmatrix},\; \begin{pmatrix} 1&0 \end{pmatrix} \times \begin{pmatrix} 0&1\\ 0&1\\ 1&0 \end{pmatrix},\; \begin{pmatrix} 1\\ 0 \end{pmatrix} \times \begin{pmatrix} 1\\ 0 \end{pmatrix}.$$