EXAMPLE B.7 Using Cramer's Rule to Solve a System of Equations

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EXAMPLE B.7 Using Cramer’s Rule to Solve a System of Equations

Use Cramer’s rule to solve the following simultaneous linear equations in three unknowns:

$$2x_1 + x_2 + x_3 = 3$$ $$x_1 + 3x_2 - x_3 = 7$$ $$x_1 + x_2 + x_3 = 1$$

In matrix form, these equations can be expressed as

⎡ 2	1	⎤ 1	⎡ x1	⎤	⎡ ⎤ 3
1 ⎣	3	−1 ⎦	x2 ⎣	⎦ =	7 ⎣ ⎦
1	1	1	x3		1

Here,

$$|\mathbf{A}| = \begin{vmatrix} 2 & 1 & 1 \\ 1 & 3 & -1 \\ 1 & 1 & 1 \end{vmatrix} = 4$$

Since |A| = 4 = 0, a unique solution exists for x1, x2, and x3. This solution is provided by Cramer’s rule [Eq. (B.21)] as follows:

$$x_1 = \frac{1}{|\mathbf{A}|} \begin{vmatrix} 3 & 1 & 1 \\ 7 & 3 & -1 \\ 1 & 1 & 1 \end{vmatrix} = \frac{8}{4} = 2$$ $$x_2 = \frac{1}{|\mathbf{A}|} \begin{vmatrix} 2 & 3 & 1 \\ 1 & 7 & -1 \\ 1 & 1 & 1 \end{vmatrix} = \frac{4}{4} = 1$$ $$x_3 = \frac{1}{|\mathbf{A}|} \begin{vmatrix} 2 & 1 & 3 \\ 1 & 3 & 7 \\ 1 & 1 & 1 \end{vmatrix} = \frac{-8}{4} = -2$$

MATLAB is well suited to compute Cramer’s formula, so these results are easy to verify. To provide an example, let us verify that x¹ = 2 using MATLAB’s det command to compute the needed matrix determinants.

>> x1 = det([3 1 1;7 3 -1;1 1 1])/det([2 1 1;1 3 -1;1 1 1])
    x1 = 2.0000