To solve the question of determining the cost of a t-shirt (x) and the cost of a notebook (y) using matrices, we will employ the method of solving linear equations using matrix operations. Given the problem, we have the following system of linear equations:
1. [tex]\(2x + 3y = 20\)[/tex]
2. [tex]\(2x + y = 8\)[/tex]
This system can also be represented in matrix form as:
[tex]\[
\left[\begin{array}{cc}
2 & 3 \\
2 & 1
\end{array}\right]
\left[\begin{array}{c}
x \\
y
\end{array}\right]
=
\left[\begin{array}{c}
20 \\
8
\end{array}\right]
\][/tex]
Here, the matrix [tex]\( \left[\begin{array}{cc} 2 & 3 \\ 2 & 1 \end{array}\right] \)[/tex] is our coefficient matrix [tex]\( A \)[/tex], the column vector [tex]\( \left[\begin{array}{c} x \\ y \end{array}\right] \)[/tex] represents the variables we need to solve for, and the column vector [tex]\( \left[\begin{array}{c} 20 \\ 8 \end{array}\right] \)[/tex] is the constants matrix [tex]\( B \)[/tex] representing the total sales.
To solve for the variables [tex]\( x \)[/tex] (the cost of a t-shirt) and [tex]\( y \)[/tex] (the cost of a notebook), we will use the inverse of the coefficient matrix [tex]\( A \)[/tex]. The steps to solve the matrix equation [tex]\( A[x, y]^T = B \)[/tex] are as follows:
1. Find the Inverse of the Coefficient Matrix [tex]\( A \)[/tex]:
The inverse of a 2x2 matrix
[tex]\[
\left[\begin{array}{cc}
a & b \\
c & d
\end{array}\right]
\][/tex]
is given by
[tex]\[
\frac{1}{ad - bc}
\left[\begin{array}{cc}
d & -b \\
-c & a
\end{array}\right]
\][/tex]
For our matrix [tex]\( A = \left[\begin{array}{cc} 2 & 3 \\ 2 & 1 \end{array}\right] \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( c = 2 \)[/tex]
- [tex]\( d = 1 \)[/tex]
First, calculate the determinant of [tex]\( A \)[/tex]:
[tex]\[
\text{det}(A) = ad - bc = (2 \cdot 1) - (3 \cdot 2) = 2 - 6 = -4
\][/tex]
Now, find the inverse of [tex]\( A \)[/tex]:
[tex]\[
A^{-1} = \frac{1}{-4}
\left[\begin{array}{cc}
1 & -3 \\
-2 & 2
\end{array}\right]
=
\left[\begin{array}{cc}
-\frac{1}{4} & \frac{3}{4} \\
\frac{1}{2} & -\frac{1}{2}
\end{array}\right]
\][/tex]
2. Multiply the Inverse Matrix [tex]\( A^{-1} \)[/tex] with Matrix [tex]\( B \)[/tex]:
To find [tex]\([x, y]\)[/tex], we perform the matrix multiplication:
[tex]\[
\left[\begin{array}{cc}
-\frac{1}{4} & \frac{3}{4} \\
\frac{1}{2} & -\frac{1}{2}
\end{array}\right]
\left[\begin{array}{c}
20 \\
8
\end{array}\right]
\][/tex]
Carry out the multiplication step-by-step:
For [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{1}{4} \cdot 20 + \frac{3}{4} \cdot 8 = -5 + 6 = 1
\][/tex]
For [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{2} \cdot 20 - \frac{1}{2} \cdot 8 = 10 - 4 = 6
\][/tex]
Therefore, the solution to the system of equations is:
- The cost of a t-shirt, [tex]\( x \)[/tex], is [tex]\( \$1 \)[/tex]
- The cost of a notebook, [tex]\( y \)[/tex], is [tex]\( \$6 \)[/tex]
So, the cost of a t-shirt is [tex]$1, and the cost of a notebook is $[/tex]6.
