Difference between revisions of "Multiplication of matrices"
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\mathbf{A}\cdot\mathbf{B}&\ne\mathbf{B}\cdot\mathbf{A} \\ | \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}\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} \\ | + | (\mathbf{A}+\mathbf{B})\cdot\mathbf{C}&=\mathbf{A}\cdot\mathbf{C}+\mathbf{B}\cdot\mathbf{C} &\text{for all l-by-m matrices } \mathbf{A},\mathbf{B} \text{ and m-by-n matrices } \mathbf{C}\\ |
| − | + | \mathbf{A}\cdot(\mathbf{B})+\mathbf{C})&=\mathbf{A}\cdot\mathbf{B}+\mathbf{A}\cdot\mathbf{C} &\text{for all l-by-m matrices } \mathbf{A} \text{ and m-by-n matrices }\mathbf{B},\mathbf{C} | |
\end{align}</math> | \end{align}</math> | ||
Revision as of 14:40, 16 May 2014
| ← Back: Addition of matrices | Overview: Matrices | Next: Matrix inversion → |
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 
with an m-by-n matrix 
is an l-by-n matrix 
. The components of the resulting matrix are comuputed as follows:
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]](/wiki/robotics/images/math/a/6/0/a60eb4b58962ac775002fcbe9dace286.png)
Some further rules for matrix multiplications are:
