Solve each exercise by using the inverse of the coeffcient matrix
to solve a system of equations.
63. Analysis of Orders  The Bread Box Bakery sells three types
of cakes, each requiring the amounts of the basic ingredients
shown in the following matrix.
Type of Cake
I   II  III
Flour (in cups)            [ 2 4 2]
Sugar (in cups)              [2 1 2 ]
Eggs                              [2 1 3]
To fill its daily orders for these three kinds of cake, the bakery
uses 72 cups of four, 48 cups of sugar, and 60 eggs.
(a)  Write a 3 by1 matrix for the amounts used daily.
(b)  Let the number of daily orders for cakes be a 3by1 matrix
X with entries x1
, x2 , and x3 . Write a matrix equation that
can be solved for X, using the given matrix and the matrix
from part (a).
(c)  Solve the equation from part (b) to find the number of
daily orders for each type of cake. show detail work for all; How do you compute the inverse of a 3×3  matrix manually for part c)

Let's break this problem down step by step, as requested.

---

### Part (a): Writing a 3 by 1 matrix for the amounts used daily

We are given the total amounts of flour, sugar, and eggs used daily:
- Flour: 72 cups
- Sugar: 48 cups
- Eggs: 60 eggs

So, the daily amounts used can be written as the following \(3 \times 1\) matrix:

\[
\mathbf{B} = 
\begin{pmatrix}
72 \\
48 \\
60
\end{pmatrix}
\]

---

### Part (b): Writing the matrix equation

We are provided with the matrix for the number of cups of each ingredient required for each type of cake. This matrix is:

\[
A = 
\begin{pmatrix}
2 & 4 & 2 \\
2 & 1 & 2 \\
2 & 1 & 3
\end{pmatrix}
\]

Let the number of orders for each type of cake be represented by the \(3 \times 1\) matrix \(X = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\), where:
- \(x_1\) is the number of Type I cakes,
- \(x_2\) is the number of Type II cakes,
- \(x_3\) is the number of Type III cakes.

The total amounts of flour, sugar, and eggs used daily must satisfy the following matrix equation:

\[
A \cdot X = B
\]

This equation can be written as:

\[
\begin{pmatrix}
2 & 4 & 2 \\
2 & 1 & 2 \\
2 & 1 & 3
\end{pmatrix}
\cdot
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
72 \\
48 \\
60
\end{pmatrix}
\]

---

### Part (c): Solving the matrix equation by finding the inverse of \(A\)

To solve this system for \(X\), we will compute the inverse of the matrix \(A\) and then multiply it by the matrix \(B\).

1. **Finding the inverse of matrix \(A\):**
   The inverse of a 3x3 matrix \(A\), denoted as \(A^{-1}\), can be calculated using the formula:

   \[
   A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)
   \]

   where:
   - \(\det(A)\) is the determinant of matrix \(A\),
   - \(\text{adj}(A)\) is the adjugate (or adjoint) of matrix \(A\), which is the transpose of the cofactor matrix.

   Let's compute these step by step.

#### Step 1: Determinant of matrix \(A\)
The determinant of a \(3 \times 3\) matrix \(A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\) is given by:

\[
\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For matrix \(A = \begin{pmatrix} 2 & 4 & 2 \\ 2 & 1 & 2 \\ 2 & 1 & 3 \end{pmatrix}\), we calculate:

\[
\det(A) = 2(1 \cdot 3 - 2 \cdot 1) - 4(2 \cdot 3 - 2 \cdot 2) + 2(2 \cdot 1 - 2 \cdot 1)
\]
\[
\det(A) = 2(3 - 2) - 4(6 - 4) + 2(2 - 2)
\]
\[
\det(A) = 2(1) - 4(2) + 2(0) = 2 - 8 = -6
\]

#### Step 2: Cofactor matrix
Now, we calculate the cofactor matrix of \(A\). The cofactor of each element is found by computing the determinant of the \(2 \times 2\) submatrices that remain after removing the row and column of the element.

- \(C_{11} = \det \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} = (1)(3) - (2)(1) = 3 - 2 = 1\)
- \(C_{12} = -\det \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix} = -(6 - 4) = -2\)
- \(C_{13} = \det \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix} = (2)(1) - (1)(2) = 2 - 2 = 0\)

- \(C_{21} = -\det \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix} = -(12 - 2) = -10\)
- \(C_{22} = \det \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix} = (2)(3) - (2)(2) = 6 - 4 = 2\)
- \(C_{23} = -\det \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix} = -(2 - 2) = 0\)

- \(C_{31} = \det \begin{pmatrix} 4 & 2 \\ 1 & 2 \end{pmatrix} = (4)(3) - (2)(1) = 6 - 2 = 4\)
- \(C_{32} = -\det \begin{pmatrix} 4 & 2 \\ 1 & 1 \end{pmatrix} = -(

Solve each exercise by using the inverse of the coefficient matrix to solve a system of equations. Analysis of Orders: The Bread Box Bakery sells three types of cakes, each requiring the amounts of the basic ingredients shown in the following matrix. Type of Cake I II III Flour (in cups) [2 4 2] Sugar (in cups) [2 1 2] Eggs [2 1 3] To fill its daily orders for these three kinds of cake, the bakery uses 72 cups of flour, 48 cups of sugar, and 60 eggs. (a) Write a 3 by 1 matrix for the amounts used daily. (b) Let the number of daily orders for cakes be a 3 by 1 matrix X with entries x1, x2, and x3. Write a matrix equation that can be solved for X, using the given matrix and the matrix from part (a). (c) Solve the equation from part (b) to find the number of daily orders for each type of cake. Show detailed work for all. How do you compute the inverse of a 3x3 matrix manually for part (c)?

Learn how to solve a system of equations using matrix inversion with a real-world problem involving cake orders at a bakery. This solution includes writing matrix equations, calculating the inverse of a 3x3 matrix, and applying it to solve for the number of cakes. Suitable for Grades 11-12 or college students.

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