Use Cramer’s rule to solve 0.3x1 + 0.52x2 + x3 = −0.01 0.5x1 + x2 + 1.9x3 = 0.67 0.1x1 + 0.3 x2 + 0.5x3 = −0.44

Question

Use Cramer’s rule to solve
[latex]0.3x_1 + 0.52x_2 + x_3 = −0.01[/latex]
[latex]0.5x_1 + x_2 + 1.9x_3 = 0.67[/latex]
[latex]0.1x_1 + 0.3 x_2 + 0.5x_3 = −0.44[/latex]

Accepted Answer

The determinant D can be evaluated as [Eq. (9.1)]:

[latex]D = a_{11} \left|\begin{matrix}a_{22} &a_{23}\ a_{32}& a_{33}\end{matrix}\right| − a_{12} \left|\begin{matrix}a_{21}& a_{23}\ a_{31}& a_{33}\end{matrix}\right| +a_{13} \left|\begin{matrix}a_{21} &a_{22}\ a_{31}& a_{32}\end{matrix}\right|\qquad (9.1)\[0.5 cm] D = 0.3 \left|\begin{matrix}1 &1.9\ 0.3& 0.5\end{matrix}\right| − 0.52 \left|\begin{matrix}0.5& 1.9\ 0.1& 0.5\end{matrix}\right| +1 \left|\begin{matrix}0.5&1\0.1& 0.3\end{matrix}\right| = - 0.0022[/latex]

The solution can be calculated as

[latex]x_1 = \frac{\left|\begin{matrix}−0.01& 0.52 &1\ 0.67 &1 &1.9\ −0.44& 0.3& 0.5\end{matrix}\right|}{−0.0022}=\frac{0.03278}{−0.0022}=-14.9[/latex]

[latex]x_2 = \frac{\left|\begin{matrix}0.3& −0.01 &1\ 0.5 &0.67& 1.9\ 0.1& −0.44 & 0.5\end{matrix}\right|}{−0.0022}=\frac{0.0649}{−0.0022}=-29.5[/latex]

[latex]x_3 = \frac{\left|\begin{matrix}0.3& 0.52 &−0.01\ 0.5& 1& 0.67\ 0.1& 0.3& −0.44\end{matrix}\right|}{−0.0022}=\frac{-0.04356}{−0.0022}=19.8[/latex]

Question 9.2: Use Cramer’s rule to solve 0.3x1 + 0.52x2 + x3 = −0.01 0.5x1...

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