Difference between revisions of "Multiplication of matrices"

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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:<br/><br/>
 
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:<br/><br/>
 
:<math>
 
:<math>
−
er
+
\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
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  \left[\begin{array}{cc}
 +
    b_{11} & b_{12} \\
 +
    b_{21} & b_{22}\\
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    b_{31} & b_{32}
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  \end{array}\right]=
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  \left[\begin{array}{cc}
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    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]
 
</math>
 
</math>

Revision as of 14:29, 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 \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]
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