We are given the system of equations:

1. $2 + 2y + z = 20$
2. $-3x - y - z = -18$
3. $x + y + 2z = 16$

To solve this system using the matrix inversion method, follow these steps:

Step 1: Convert the system of equations into matrix form

First, rewrite the equations in standard form (all variables on the left, constants on the right):

1. $0x + 2y + z = 18$ (Subtract 2 from both sides)
2. $-3x - y - z = -18$
3. $x + y + 2z = 16$

This system can be written as:

0 & 2 & 1 \\ -3 & -1 & -1 \\ 1 & 1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 18 \\ -18 \\ 16 \end{bmatrix}$$ Let: - Matrix $$A = \begin{bmatrix} 0 & 2 & 1 \\ -3 & -1 & -1 \\ 1 & 1 & 2 \end{bmatrix}$$ - Vector $$\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$$ - Vector $$\mathbf{b} = \begin{bmatrix} 18 \\ -18 \\ 16 \end{bmatrix}$$ Thus, the system becomes $$A \mathbf{x} = \mathbf{b}$$. ### Step 2: Find the inverse of matrix $$A$$ To solve for $$\mathbf{x}$$, we compute $$\mathbf{x} = A^{-1} \mathbf{b}$$, where $$A^{-1}$$ is the inverse of matrix $$A$$. Let’s calculate the inverse of matrix $$A$$. $$A = \begin{bmatrix} 0 & 2 & 1 \\ -3 & -1 & -1 \\ 1 & 1 & 2 \end{bmatrix}$$ To find $$A^{-1}$$, we can use methods like the cofactor matrix and adjugate or use a calculator for this step. Let me compute that for you. The solution to the system of equations is: $$x = 2.5, \quad y = 7.5, \quad z = 3$$ ### Summary: Using the matrix inversion method, we found that: - $$x = 2.5$$ - $$y = 7.5$$ - $$z = 3$$ Would you like more details on any step or further clarification? Here are 5 related questions: 1. How do you compute the determinant of a matrix? 2. What are the conditions for a matrix to have an inverse? 3. How can the Gaussian elimination method be used to solve a system of equations? 4. What is the cofactor method for finding a matrix inverse? 5. How can you verify if the solution of a system is correct? **Tip:** A matrix is invertible (non-singular) if its determinant is non-zero.