Question

Using matrix method, solve the simultaneous equations

{x-3y=3

5x-9y=11

Answer 1

$$AX=B$$

$$A^{-1}AX=A^{-1}B$$

$$X=A^{-1}B$$

$$A=\begin{pmatrix} 1 & -3 \\ 5 & -9 \end{pmatrix}, X=\begin{pmatrix} x \\ y \end{pmatrix},B=\begin{pmatrix} 3 \\ 11 \end{pmatrix}$$

$$\begin{pmatrix} 1 & -3 \\ 5 & -9 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} 3 \\ 11 \end{pmatrix}$$

$$\det A=\begin{vmatrix} 1 & -3 \\ 5 & -9 \end{vmatrix}=1(-9)-(-3)(5)=6\not=0$$

$$A^{-1}=\dfrac{1}{6}\begin{pmatrix} -9 & 3 \\ -5 & 1 \end{pmatrix}=\begin{pmatrix} -3/2 & 1/2 \\ -5/6 & 1/6 \end{pmatrix}$$

$$\begin{pmatrix} -3/2 & 1/2 \\ -5/6 & 1/6 \end{pmatrix}\begin{pmatrix} 3\\ 11 \end{pmatrix}=\begin{pmatrix} -9/2+11/2 \\ -15/6+11/6 \end{pmatrix}$$

$$=\begin{pmatrix} 1 \\ -2/3 \end{pmatrix}$$

$$x=1, y=-\dfrac{2}{3}$$

$$(1, -\dfrac{2}{3})$$