Matrix Equations

In general a matrix

$\displaystyle A = \left[\begin{array}{cccc} a_{11} & a_{12} & \cdots\cdots & a_...
...\vdots & & \vdots  a_{m1} & a_{m2} & \cdots\cdots & a_{mn} \end{array}\right]$ (2)

consists of $ m$ rows and $ n$ columns, and is called an $ m\times n$ matrix. For example, $ \left[\begin{array}{ccc}
1 & 2 & -1 \\
0 & -5 & 3
\end{array}\right]$ is a $ 2\times 3$ matrix.

Now consider the above matrix as coefficient. Then the product $ A \mathbf{x}$ of an $ m\times n$ matrix with a $ n$-dimensional vector is given by

$\displaystyle \left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots\cdots & a_{1n}...
...s\cdots \\
a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n
\end{array}\right] ,
$

and results in an $ m$-dimensional vector. With another $ m$-dimensional vector $ \mathbf{b}$ we can express a system of linear equations as follows.

$\displaystyle \left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots\cdots & a_{1n}...
...ray}\right]
=
\left[\begin{array}{c}
b_1 \\
\vdots \\
b_m
\end{array}\right]
$

Or, we can simply write $ A \mathbf{x} = \mathbf{b}$.

Provided a $ m\times n$ matrix A and a vector u in $ \mathbb{R}^n$, we can compute the multiplication as follows.

> A * u

EXAMPLE 1. Compute $ A \mathbf{u}$ in each of the following.

  1. $ A =
\left[\begin{array}{ccc}
1 & 2 & -1 \\
0 &-5 & 3
\end{array}\right]$    and $ \quad
\mathbf{u} =
\left[\begin{array}{c}
4 \\
3 \\
7
\end{array}\right]$

  2. $ A =
\left[\begin{array}{cc}
2 &-3 \\
8 & 0 \\
-5 & 2
\end{array}\right]$    and $ \quad
\mathbf{u} =
\left[\begin{array}{c}
4 \\
7
\end{array}\right]$



Subsections

Department of Mathematics
Last modified: 2005-10-21