Matrix operations

SCALAR MULTIPLE: Given any matrix $ A$, we can define the scalar multiple $ c A$ with scalar $ c$. For example,

$\displaystyle c \;
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22}...
...times a_{13} \\
c\times a_{21} & c\times a_{22} & c\times a_{23}
\end{bmatrix}$

SUM: Given two $ m\times n$ matrices $ A$ and $ B$, we can define the sum $ A + B$ of $ m\times n$ matrix. For example,

$\displaystyle \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_...
... + b_{13} \\
a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23}
\end{bmatrix}$

MATRIX MULTIPLICATION: Given $ l\times m$ matrix $ A$ and $ m\times n$ matrix $ B$, we can define the multiplication $ A B$ of $ l\times n$ matrix. Suppose, for example, that we have a $ 2\times 3$ matrix $ A$ and a $ 3\times 2$ matrix $ B$. Then we can compute the multiplication $ A B$ of $ 2 \times 2$ matrix by

$\displaystyle \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_...
...} + a_{23} b_{31}
& a_{21} b_{12} + a_{22} b_{22} + a_{23} b_{32}
\end{bmatrix}$

The matrix multiplication is not commutative. That is, $ A B \neq B A$ in general.



Subsections

Department of Mathematics
Last modified: 2005-10-21