The inverse matrix associated to the system of linear equations described in this question is equal to the matrix [tex]\vec {A}^{-1} = \left[\begin{array}{ccc}-19&9&-7\\15&-7&6\\-2&1&-1\end{array}\right][/tex]. (Correct choice: B)
How to determine the inverse matrix associated to a system of linear equations
A system of linear equations have an unique solution when the number of variables is equal to the number of linear equations. There are several ways to solve a system of three linear equations with three variables, one approach consists in using the concepts of operations between matrices and inverse matrix, for a linear system of the form [tex]\vec A \cdot \vec x = \vec B[/tex] it follows a solution of the form:
[tex]\vec x = \vec {A}^{-1} \cdot \vec B[/tex] (1)
Where:
- [tex]\vec A[/tex] - Matrix of dependent constants.
- [tex]\vec B[/tex] - Matrix of independent constants.
- [tex]\vec{A}^{-1}[/tex] - Inverse matrix of dependent constants.
- [tex]\vec x[/tex] - Solution matrix.
And the inverse of the dependent constants is determined by the following expression:
[tex]\vec {A}^{-1} = \frac{adj (\vec A)}{\det(\vec A)}[/tex] (2)
Where:
- [tex]adj(\vec A)[/tex] - Adjugate of the matrix of dependent constants.
- [tex]\det (\vec A)[/tex] - Determinant of the matrix of dependent constants.
Please notice that the adjugate is the matrix of cofactors of a given matrix.
By applying the concepts of determinant and adjugate we have the following results:
[tex]\det (\vec A) = 1[/tex]
[tex]adj(\vec A) = \left[\begin{array}{ccc}-19&9&-7\\15&-7&6\\-2&1&-1\end{array}\right][/tex]
[tex]\vec {A}^{-1} = \left[\begin{array}{ccc}-19&9&-7\\15&-7&6\\-2&1&-1\end{array}\right][/tex]
The inverse matrix associated to the system of linear equations described in this question is equal to the matrix [tex]\vec {A}^{-1} = \left[\begin{array}{ccc}-19&9&-7\\15&-7&6\\-2&1&-1\end{array}\right][/tex]. (Correct choice: B)
Remark
The statement of the question is poorly formatted. Correct form is shown below:
What is the inverse matrix that can be used to solve this system of equations?
x + 2 · y + 5 · z = 14
3 · x + 5 · y + 9 · z = -1
x + y - 2 · z = 6
To learn more on inverse matrices: https://brainly.com/question/4017205
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