Using matrix inversion method, solve the system of equations: x + 2y - 6z = 5, 2x - 4y + 5z = 6, 3x - 16y + 9z = 4.

To solve the given system of equations using the matrix inversion method, follow these steps:

### System of Equations:
\[
\begin{aligned}
1. & \quad x + 2y - 6z = 5 \\
2. & \quad 2x - 4y + 5z = 6 \\
3. & \quad 3x - 16y + 9z = 4
\end{aligned}
\]

### Step 1: Express the system in matrix form
The system can be written as \( A \mathbf{x} = \mathbf{b} \), where:
\[
A = \begin{pmatrix}
1 & 2 & -6 \\
2 & -4 & 5 \\
3 & -16 & 9
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix}
5 \\
6 \\
4
\end{pmatrix}
\]

### Step 2: Find the inverse of matrix \( A \)
The inverse of a matrix \( A \), denoted \( A^{-1} \), is calculated as follows:
\[
A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)
\]
where \( \text{det}(A) \) is the determinant of \( A \), and \( \text{adj}(A) \) is the adjugate of \( A \).

#### Calculate \( \text{det}(A) \):
\[
\text{det}(A) = 1 \times \begin{vmatrix} -4 & 5 \\ -16 & 9 \end{vmatrix} - 2 \times \begin{vmatrix} 2 & 5 \\ 3 & 9 \end{vmatrix} + (-6) \times \begin{vmatrix} 2 & -4 \\ 3 & -16 \end{vmatrix}
\]

\[
\text{det}(A) = 1 \times ((-4)(9) - (5)(-16)) - 2 \times ((2)(9) - (5)(3)) - 6 \times ((2)(-16) - (-4)(3))
\]

\[
\text{det}(A) = 1 \times (-36 + 80) - 2 \times (18 - 15) - 6 \times (-32 + 12)
\]

\[
\text{det}(A) = 1 \times 44 - 2 \times 3 - 6 \times (-20)
\]

\[
\text{det}(A) = 44 - 6 + 120 = 158
\]

#### Calculate \( \text{adj}(A) \):
To calculate the adjugate matrix \( \text{adj}(A) \), compute the cofactor matrix and then take its transpose.

### Step 3: Compute \( \mathbf{x} = A^{-1} \mathbf{b} \)
Once we have \( A^{-1} \), multiply it by the vector \( \mathbf{b} \) to find \( \mathbf{x} \).

Let's perform these calculations step by step.

#### Step 2: Finding the Inverse Matrix

Let me calculate the inverse matrix \( A^{-1} \) and then solve for \( \mathbf{x} \).The solution to the system of equations is:

\[
x = 4, \quad y = 0.5, \quad z = 0
\]

This means the values of \(x\), \(y\), and \(z\) that satisfy all three equations are \( x = 4 \), \( y = 0.5 \), and \( z = 0 \).

Would you like more details or have any questions?

### Related Questions:
1. How do you compute the determinant of a 3x3 matrix?
2. What is the adjugate of a matrix, and how is it used in matrix inversion?
3. Can you solve a system of linear equations if the determinant is zero?
4. What are other methods to solve a system of linear equations besides matrix inversion?
5. How does matrix multiplication work in solving systems of equations?
6. What are the applications of matrix inversion in real-world problems?
7. How does numerical precision affect the accuracy of matrix inversion?
8. What is the significance of the inverse matrix in linear algebra?

### Tip:
When solving systems of equations using matrices, ensure that the determinant is not zero, as this would mean the matrix is singular and doesn't have an inverse.

Learn how to solve a system of equations using the matrix inversion method step-by-step. This problem covers linear algebra concepts including matrix inversion, determinant calculation, and systems of equations. Suitable for advanced high school students and college-level learners.

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