Respuesta :
Answer:
B. z = 2
Step-by-step explanation:
Set up the matrix like this. I multiplied the last row by a 2 to get rid of the fractions so that's what you will see:
[tex]\left[\begin{array}{ccc}2&1&-1\\0&2&3\\-1&2&2\end{array}\right][/tex]
I used Cramer's Rule to solve for the value of y. In order to do that you need to find the determinant of the matrix, then you need to find the determinant of the matrix after you sub the solution set into the column representing z.
Find the determinant requires that I "pick up" the first 2 columns and then drop them at the end of the matrix and do the multiplication of the majors minus the minors. The matrix looks like this:
2 1 -1 2 1
0 2 3 0 2
-1 2 2 -1 2
The multiplication of the majors:
[(2×2×2)+(1×3×-1)+(-1×0×2)] = (8-3+0)=5
The multiplication of the minors:
[(-1×2×-1)+(2×3×2)+(2×0×1)] = (2+12+0) = 14
So the determinant of the matrix is I A I = 5 - 14 = -9
Now for the determinant of z, noted as I [tex]A_{z}[/tex] I. Notice that I am replacing the laast column with the solution set this time:
2 1 -8 2 1
0 2 -6 0 2
-1 2 -8 -1 2
The multiplication of the majors:
[(2×2×-8)+(1×-6×-1)+(-8×0×2)] = (-32+6+0) = -26
The multiplication of the minors:
[(-1×2×-8)+(2×-6×2)+(-8×0×1)] = (16 - 24 + 0) = -8
So the determinant of I [tex]A_{z}[/tex] I = -18
Cramer's Rule is to divide I [tex]A_{z}[/tex] I by I A I:
[tex]\frac{-18}{-9}= 2[/tex]
So the value you need for z to solve for y is 2
The value of z is 2.
The correct answer is an option (B) z = 2
What is a matrix?
"It is a set of numbers arranged in rows and columns so as to form a rectangular array."
What is inverse of matrix?
"An inverse of matrix 'B' is a matrix such that [tex]B\times B'=I[/tex] where [tex]I[/tex] is an identity matrix."
What is system of equations?
"It is a set of equations for which we find a common solution."
For given question,
We have been given a system of equations.
2x + y - z = -8
2y + 3z = -6
-1/2x + y + z = -4
We can write the third equation as, -x + 2y + 2z = -8
So, we have system of equations,
2x + y - z = -8
2y + 3z = -6
-x + 2y + 2z = -8
We write above system of equations in matrix form as,
[tex]\Rightarrow \begin{bmatrix}2 & 1 & -1 \\0 & 2 & 3 \\-1 & 2 & 2\end{bmatrix}[/tex] [tex]\begin{bmatrix}x \\y \\z\end{bmatrix}[/tex] [tex]=\begin{bmatrix}-8 \\-6 \\-8\end{bmatrix}[/tex]
⇒ AX = B
We use inverse matrix to find the solution of given system of equations.
Pre-multiply both the sides of above equation by [tex]A^{-1}[/tex]
[tex]\Rightarrow A^{-1} AX = A^{-1}B\\\\\Rightarrow IX=A^{-1}B\\\\\Rightarrow X=A^{-1}B[/tex]
The inverse of matrix A is,
[tex]A^{-1}=\begin{bmatrix}\frac{2}{9} & \frac{4}{9} & \frac{-5}{9} \\\\\frac{1}{3} & \frac{-1}{3} & \frac{2}{3} \\\\\frac{-2}{9} & \frac{5}{9} & \frac{-4}{9}\end{bmatrix}[/tex]
So, the solution of given system of equations would be,
[tex]\Rightarrow \begin{bmatrix}x \\y \\z\end{bmatrix}=\begin{bmatrix}\frac{2}{9} & \frac{4}{9} & \frac{-5}{9} \\\\\frac{1}{3} & \frac{-1}{3} & \frac{2}{3} \\\\\frac{-2}{9} & \frac{5}{9} & \frac{-4}{9}\end{bmatrix} . \begin{bmatrix}-8 \\-6 \\-8\end{bmatrix}[/tex]
[tex]\Rightarrow \begin{bmatrix}x \\y \\z\end{bmatrix}= \begin{bmatrix}0 \\-6 \\2\end{bmatrix}[/tex]
Therefore, the value of z is 2.
The correct answer is an option (B) z = 2
Learn more about the system of equations here:
https://brainly.com/question/22798746
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