https://file.aiquickdraw.com/mathbot/file/1727403298323bqpc2la7.jpg

Let's break down the problem and solve it step by step:

We are given the matrix \( A \) and asked to perform three tasks:

### Matrix A:
\[
A = \begin{pmatrix}
2 & 6 & 6 \\
4 & 14 & 13 \\
2 & 7 & 7
\end{pmatrix}
\]

### Part (a): Use the inversion algorithm to find \( A^{-1} \)

To find \( A^{-1} \), we will use the formula for the inverse of a matrix:
\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]
1. **Step 1: Find \( \text{det}(A) \)**  
2. **Step 2: Find the adjugate matrix \( \text{adj}(A) \)**  
3. **Step 3: Calculate \( A^{-1} \) using the formula**

### Part (b): Express \( A\mathbf{x} = \mathbf{b} \) as a linear system

Given:
\[
\mathbf{x} = \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix}
1 \\
-3 \\
2
\end{pmatrix}
\]

This gives us the system of linear equations:
\[
\begin{aligned}
2x_1 + 6x_2 + 6x_3 &= 1 \\
4x_1 + 14x_2 + 13x_3 &= -3 \\
2x_1 + 7x_2 + 7x_3 &= 2
\end{aligned}
\]

### Part (c): Use \( A^{-1} \) to solve the system

Once we have \( A^{-1} \), we will solve for \( \mathbf{x} \) using the relation:
\[
\mathbf{x} = A^{-1} \mathbf{b}
\]

Now, I will calculate the determinant of \( A \), find the adjugate, and determine \( A^{-1} \). Let's proceed with that calculation.### Part (a): Inverse of Matrix \( A \)

The determinant of \( A \) is \( \text{det}(A) = 2 \), which is non-zero, so the inverse exists. The inverse matrix \( A^{-1} \) is:

\[
A^{-1} = \begin{pmatrix}
3.5 & 0 & -3 \\
-1 & 1 & -1 \\
0 & -1 & 2
\end{pmatrix}
\]

### Part (b): Linear System

The system of equations \( A\mathbf{x} = \mathbf{b} \) is expressed as:
\[
\begin{aligned}
2x_1 + 6x_2 + 6x_3 &= 1 \\
4x_1 + 14x_2 + 13x_3 &= -3 \\
2x_1 + 7x_2 + 7x_3 &= 2
\end{aligned}
\]

### Part (c): Solving the System Using \( A^{-1} \)

Using the inverse matrix \( A^{-1} \), we solve for \( \mathbf{x} \) by calculating \( \mathbf{x} = A^{-1} \mathbf{b} \). The solution is:

\[
\mathbf{x} = \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix} = \begin{pmatrix}
-2.5 \\
-6 \\
7
\end{pmatrix}
\]

This is the solution to the system.

---

Would you like further details or have any questions?

Here are some related questions you could explore:
1. How is the adjugate matrix \( \text{adj}(A) \) calculated?
2. What happens if the determinant of a matrix is zero?
3. How does matrix inversion differ from solving systems with Gaussian elimination?
4. Can you solve a system with dependent or inconsistent equations using inverses?
5. How does matrix inversion behave computationally for larger matrices?

**Tip:** The determinant gives you insight into whether a matrix is invertible, but it also reveals important geometric properties like the volume scaling factor of linear transformations.

Consider the matrix A = [[2, 6, 6], [4, 14, 13], [2, 7, 7]]. (a) Use the inversion algorithm to find A^(-1). (b) Express Ax = b as a linear system where x = (x1, x2, x3) and b = (1, -3, 2). (c) Use A^(-1) to solve the linear system in part (b).

Learn how to find the inverse of a 3x3 matrix A and solve the system of equations Ax = b, where x = (x1, x2, x3) and b = (1, -3, 2). This problem involves key concepts in matrix algebra and linear systems, suitable for undergraduate students.

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