Multiplication of matrices

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Two matrices can be multiplied if the number of colums of the left matrix equals the number of rows of the right matrix. The result of the multiplication of an l-by-m matrix $\mathbf{A}=(a_{ij})_{i=1...l,j=1...m}$ with an m-by-n matrix $\mathbf{B}=(b_{ij})_{i=1...m,j=1...n}$ is an l-by-n matrix $\mathbf{C}=(c_{ij})_{i=1...l,j=1...n}$. The components of the resulting matrix are comuputed as follows:

$c_{ij}=\sum^{m}_{k=1}a_{ik}\cdot b_{kj}$

For example the multiplication of a 2-by-3 matrix with a 3-by-2 matrix results in a 2-by-2 matrix and is computed as follows:

$\mathbf{A}\cdot\mathbf{B}= \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{array}\right]\cdot \left[\begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22}\\ b_{31} & b_{32} \end{array}\right]= \left[\begin{array}{cc} a_{11}\cdot b_{11}+a_{12}\cdot b_{21}+a_{13}\cdot b_{31} & a_{11}\cdot b_{12}+a_{12}\cdot b_{22}+a_{13}\cdot b_{12} \\ a_{21}\cdot b_{11}+a_{22}\cdot b_{21}+a_{23}\cdot b_{31} & a_{21}\cdot b_{12}+a_{22}\cdot b_{22}+a_{23}\cdot b_{12} \end{array}\right]$

Some further rules for matrix multiplications are:

$\begin{align} \mathbf{A}\cdot\mathbf{B}&\ne\mathbf{B}\cdot\mathbf{A} \\ (\mathbf{A}\cdot\mathbf{B})\cdot\mathbf{C}&=\mathbf{A}\cdot(\mathbf{B}\cdot\mathbf{A}\cdot\mathbf{C}) \\ (\mathbf{A}+\mathbf{B})\cdot\mathbf{C}&=\mathbf{A}\cdot\mathbf{C}+\mathbf{B}\cdot\mathbf{C} \\ \end{align}$