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What is the inverse matrix that can be used to solve this system of equations?
x + 2y + 5z = 14
3x + 5y + 9z = −1
x + y
= 2z = 6
-
O A.
P
-19 9
-2 1
15 -7 6
OB.
T-19 9 -7
15 -7
-2
$3
6
O C.
O D.
0 0
010
L0 0 1J

What Is The Inverse Matrix That Can Be Used To Solve This System Of Equations X 2y 5z 14 3x 5y 9z 1 X Y 2z 6 O A P 19 9 2 1 15 7 6 OB T19 9 7 15 7 2 3 6 O C O D class=

Sagot :

xero099

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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