Answered

Select the correct matrices.

Winston pays [tex]$8$[/tex] for a burger, an order of fries, and a soft drink. Tia buys 2 burgers and a soft drink for [tex]$10.50. George buys 2 orders of fries, a burger, and 2 soft drinks for $[/tex]12[tex]$. Let $[/tex]x, y,[tex]$ and $[/tex]z[tex]$ represent the cost of a burger, an order of fries, and a soft drink, respectively. If $[/tex]A[tex]$ is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, $[/tex]A^{-1}[tex]$, and the solution matrix, $[/tex]X$, of the system of equations.

[tex]\[
\begin{array}{c}
\left[\begin{array}{rrr}
2 & 1 & -1 \\
3 & -1 & -1 \\
-4 & 1 & 3
\end{array}\right]
\left[\begin{array}{c}
4 \\
2.5 \\
1.5
\end{array}\right]
\left[\begin{array}{rrr}
2 & 0 & -1 \\
3 & -1 & -1 \\
-4 & 1 & 2
\end{array}\right]
\left[\begin{array}{c}
3.5 \\
2 \\
2.5
\end{array}\right] \\
\left[\begin{array}{c}
4.5 \\
1.5 \\
2
\end{array}\right]
\left[\begin{array}{rrr}
2 & 0 & -1 \\
3 & -2 & -1 \\
-4 & 1 & 3
\end{array}\right]
\left[\begin{array}{c}
4 \\
1.5 \\
2.5
\end{array}\right]
\left[\begin{array}{rrr}
2 & 1 & -1 \\
3 & -2 & -1 \\
-4 & 1 & 2
\end{array}\right]
\end{array}
\][/tex]



Answer :

GinnyAnswer
To identify the correct matrices, we need to accurately represent the given system of equations and their respective matrices.

The equations based on the problem description are:

1. [tex]\( x + y + z = 8 \)[/tex]
2. [tex]\( 2x + z = 10.5 \)[/tex]
3. [tex]\( x + 2y + 2z = 12 \)[/tex]

Here, [tex]\( x \)[/tex] represents the cost of a burger, [tex]\( y \)[/tex] the cost of an order of fries, and [tex]\( z \)[/tex] the cost of a soft drink.

Given these equations, let's write them in matrix form:

- The coefficient matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 0 & 1 \\ 1 & 2 & 2 \end{bmatrix} \][/tex]

- The result matrix [tex]\( B \)[/tex]:
[tex]\[ B = \begin{bmatrix} 8 \\ 10.5 \\ 12 \end{bmatrix} \][/tex]

Next, we need to find the inverse matrix [tex]\( A^{-1} \)[/tex] and the solution matrix [tex]\( X \)[/tex].

Using the matrix [tex]\( A \)[/tex], its inverse [tex]\( A^{-1} \)[/tex] is:
[tex]\[ A^{-1} = \begin{bmatrix} 2 & 0 & -1 \\ 3 & -1 & -1 \\ -4 & 1 & 2 \end{bmatrix} \][/tex]

The solution matrix [tex]\( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \)[/tex] can be found by [tex]\( X = A^{-1}B \)[/tex]:
[tex]\[ X = \begin{bmatrix} 4 \\ 1.5 \\ 2.5 \end{bmatrix} \][/tex]

Therefore, the correct matrices from the given options are:

- The inverse matrix [tex]\( A^{-1} \)[/tex]:
[tex]\[ \begin{bmatrix} 2 & 0 & -1 \\ 3 & -1 & -1 \\ -4 & 1 & 2 \end{bmatrix} \][/tex]

- The solution matrix [tex]\( X \)[/tex]:
[tex]\[ \begin{bmatrix} 4 \\ 1.5 \\ 2.5 \end{bmatrix} \][/tex]

So the correct pair of matrices from the options given are:
[tex]\[ \left[\begin{array}{rrr} 2 & 0 & -1 \\ 3 & -1 & -1 \\ -4 & 1 & 2 \end{array}\right], \quad \left[\begin{array}{c} 4 \\ 1.5 \\ 2.5 \end{array}\right] \][/tex]