The solution of the system of linear equations by using the solving method based on the inverse matrix method is x ≈ 0.032, y ≈ 2.458 and z ≈ 1.531.
How to solve a system of linear equations by using matrices
Any system of linear equations can be described by the following linear algebra formula:
[tex]\vec A \cdot \vec X = \vec B[/tex] (1)
Where:
- [tex]\vec A[/tex] - Matrix of dependent coefficients.
- [tex]\vec X[/tex] - Vector column of variables.
- [tex]\vec B[/tex] - Vector column of independent coefficients.
A solution exists if and only if the determinant of the matrix of dependent coefficients does not equals to zero, that is to say:
[tex]det(\vec A) \neq 0[/tex] (2)
If (2) is observed, then the solution of the system is defined by:
[tex]\vec X = (\vec A)^{-1}\cdot \vec B[/tex] (3)
Where [tex](\vec A)^{-1}[/tex] is the inverse of the matrix of dependent coefficients.
Now we proceed to solve the system given:
[tex]\det (\vec A) =-123[/tex]
[tex]\left[\begin{array}{c}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}\frac{12}{41} &-\frac{1}{41} &-\frac{9}{41} \\-\frac{7}{123} &\frac{4}{123} &\frac{12}{41} \\-\frac{28}{123} &\frac{16}{123} &\frac{7}{41} \end{array}\right]\cdot \left[\begin{array}{c}7.5\\14\\8.3\end{array}\right][/tex]
[tex]\left[\begin{array}{c}x\\y\\z\end{array}\right] = \left[\begin{array}{c}\frac{13}{410} \\\frac{3023}{1230} \\\frac{1883}{1230} \end{array}\right][/tex]
Therefore, the solution of the system of linear equations by using the solving method based on the inverse matrix method is x ≈ 0.032, y ≈ 2.458 and z ≈ 1.531.
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