The problem provides a system of three linear equations with three unknowns ($e$, $h$, and $n$), representing the costs of Earrings, Headbands, and Necklaces, respectively. The task is to identify the correct matrix representation of the solution to this system of equations from the given options (A, B, C, D). Then, demonstrate that the chosen option is correct. The equations are: $2e + 2h = 8$ $4h + 2n = 5$ $4e + 2h + 4n = 15$

1. Problem Description

The problem provides a system of three linear equations with three unknowns ($e$, $h$, and $n$), representing the costs of Earrings, Headbands, and Necklaces, respectively. The task is to identify the correct matrix representation of the solution to this system of equations from the given options (A, B, C, D). Then, demonstrate that the chosen option is correct. The equations are:

$2e + 2h = 8$

$4h + 2n = 5$

$4e + 2h + 4n = 15$

2. Solution Steps

First, rewrite the system of linear equations in matrix form $Ax = b$:

$$

\begin{bmatrix}

2 & 2 & 0 \\

0 & 4 & 2 \\

4 & 2 & 4

\end{bmatrix}

\begin{bmatrix}

e \\

h \\

n

\end{bmatrix}

=

\begin{bmatrix}

8 \\

5 \\

15

\end{bmatrix}

$$

The solution to the system of linear equations is given by $x = A^{-1}b$. Therefore,

$$

\begin{bmatrix}

e \\

h \\

n

\end{bmatrix}

=

\begin{bmatrix}

2 & 2 & 0 \\

0 & 4 & 2 \\

4 & 2 & 4

\end{bmatrix}^{-1}

\begin{bmatrix}

8 \\

5 \\

15

\end{bmatrix}

$$

Comparing this with the given options, options A and C have the correct form $x = A^{-1}b$, whereas B has the form $x = Ab$, and D has a different matrix. Since the second column of the matrix should consist of $\{2, 4, 2\}$, C is not correct. Comparing A and D:

A:

$$

\begin{bmatrix}

e \\

h \\

n

\end{bmatrix}

=

\begin{bmatrix}

2 & 2 & 0 \\

0 & 4 & 2 \\

4 & 2 & 4

\end{bmatrix}^{-1}

\begin{bmatrix}

8 \\

5 \\

15

\end{bmatrix}

$$

D:

$$

\begin{bmatrix}

e \\

h \\

n

\end{bmatrix}

=

\begin{bmatrix}

2 & 0 & 2 \\

4 & 2 & 0 \\

4 & 2 & 4

\end{bmatrix}^{-1}

\begin{bmatrix}

8 \\

5 \\

15

\end{bmatrix}

$$

Now let's calculate $A^{-1}$ of

$$

A = \begin{bmatrix}

2 & 2 & 0 \\

0 & 4 & 2 \\

4 & 2 & 4

\end{bmatrix}

$$

First calculate the determinant of A:

$det(A) = 2(4*4 - 2*2) - 2(0*4 - 2*4) + 0(0*2 - 4*4) = 2(16 - 4) - 2(-8) = 24 + 16 = 40$

Next, calculate the matrix of cofactors:

$$

C = \begin{bmatrix}

12 & 8 & -16 \\

-8 & 8 & 4 \\

4 & -4 & 8

\end{bmatrix}

$$

Then, calculate the adjugate (transpose of the cofactor matrix):

$$

adj(A) = \begin{bmatrix}

12 & -8 & 4 \\

8 & 8 & -4 \\

-16 & 4 & 8

\end{bmatrix}

$$

Finally, calculate the inverse:

$$

A^{-1} = \frac{1}{40} \begin{bmatrix}

12 & -8 & 4 \\

8 & 8 & -4 \\

-16 & 4 & 8

\end{bmatrix}

= \begin{bmatrix}

3/10 & -1/5 & 1/10 \\

1/5 & 1/5 & -1/10 \\

-2/5 & 1/10 & 1/5

\end{bmatrix}

$$

Now let us solve for e, h, and n:

$$

\begin{bmatrix}

e \\

h \\

n

\end{bmatrix}

= \begin{bmatrix}

3/10 & -1/5 & 1/10 \\

1/5 & 1/5 & -1/10 \\

-2/5 & 1/10 & 1/5

\end{bmatrix}

\begin{bmatrix}

8 \\

5 \\

15

\end{bmatrix}

$$

$e = (3/10)*8 + (-1/5)*5 + (1/10)*15 = 24/10 - 5/5 + 15/10 = 2.4 - 1 + 1.5 = 2.9$

$h = (1/5)*8 + (1/5)*5 + (-1/10)*15 = 8/5 + 5/5 - 15/10 = 1.6 + 1 - 1.5 = 1.1$

$n = (-2/5)*8 + (1/10)*5 + (1/5)*15 = -16/5 + 5/10 + 15/5 = -3.2 + 0.5 + 3 = 0.3$

Now let us verify if those values work with the original equations:

$2e + 2h = 2*2.9 + 2*1.1 = 5.8 + 2.2 = 8$

$4h + 2n = 4*1.1 + 2*0.3 = 4.4 + 0.6 = 5$

$4e + 2h + 4n = 4*2.9 + 2*1.1 + 4*0.3 = 11.6 + 2.2 + 1.2 = 15$

Therefore, the option A is the correct option.

3. Final Answer

A.

$$

\begin{bmatrix}

e \\

h \\

n

\end{bmatrix}

=

\begin{bmatrix}

2 & 2 & 0 \\

0 & 4 & 2 \\

4 & 2 & 4

\end{bmatrix}^{-1}

\begin{bmatrix}

8 \\

5 \\

15

\end{bmatrix}

$$