4 Equations in Matrix Algebra

Matrix algebra provides a very convenient short hand for writing sets of equations. For example, the pair of simultaneous equations

$$\begin{eqnarray*} y_{1}&=&2x_{1} + 3x_{2}\\ y_{2}&=&x_{1} + x_{2} \end{eqnarray*}$$

may be written

$$\begin{eqnarray*} {\bf y} & = & {\bf Ax} \end{eqnarray*}$$

i.e.,

$$\begin{eqnarray*} \left( \begin{array}{r} y_{1} \ y_{2} \end{array} \right) &... ...ht) \left( \begin{array}{r} x_{1} \ x_{2} \end{array} \right) \end{eqnarray*}$$

Also if we have the following pair of equations:

$$\begin{eqnarray*} {\bf y} &=& {\bf Ax}\\ {\bf x} &=& {\bf Bz}, \end{eqnarray*}$$

then

$$\begin{eqnarray*} {\bf y} &=& \bf {A(Bz)}\\ &=& \bf {ABz}\\ &=& \bf {Cz} \end{eqnarray*}$$

where ${\bf C=AB}$. This is very convenient notation compared with direct substitution. The Mx structural equations are written in this general form, i.e.,

Real variables (y) = Matrix $\times$ hypothetical variables.

To show the simplicity of the matrix notation, consider the following equations:

$$\begin{eqnarray*} y_{1}&=&2x_{1} + 3x_{2}\\ y_{2}&=&x_{1} + x_{2}\\ x_{1}&=&z_{1} + z_{2}\\ x_{2}&=&z_{1} - z_{2} \end{eqnarray*}$$

Then we have

$$\begin{eqnarray*} y_{1}&=&2(z_{1}+z_{2}) + 3(z_{1}-z_{2})\\ &=& 5z_{1}-z_{2}\\ y_{2}&=&(z_{1}+z_{2}) + (z_{1}-z_{2}) \\ &=& 2z_{1} + 0 \end{eqnarray*}$$

Similarly, in matrix notation, we have ${\bf y=ABz}$, where

$${\bf A} = \left( \begin{array}{rr} 2 & 3\ 1 & 1 \end{array} ... ... \left( \begin{array}{rr} 1 & 1 \ 1 & -1 \end{array} \right)$$

and

$${\bf AB} = \left( \begin{array}{rr} 5 & -1\ 2 & 0 \end{array} \right), \;$$

or

$$\begin{eqnarray*} y_{1}&=&5z_{1} - z_{2}\\ y_{2}&=&2z_{2} \end{eqnarray*}$$