To find the inverse of matrix $A$ and solve the system $Ax = b$, we need to perform the following steps:

Step 1: Define the Matrix $A$ and Vector $b$

The given matrix $A$ is:

$A = \begin{bmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ 1 & -2 & 0 \end{bmatrix}$

And the vector $b$ is:

$b = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$

Step 2: Find the Inverse of Matrix $A$

The inverse of a matrix $A$, denoted as $A^{-1}$, can be found using the formula:

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

where:

• $\text{det}(A)$ 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 of $A$.

Step 3: Solve the System $Ax = b$

Once we find $A^{-1}$, we can solve for $x$ using:

$x = A^{-1}b$

Let's compute the inverse of matrix $A$ and solve for $x$.### Inverse of Matrix $A$

The inverse of matrix $A$ is:

-0.4444 & 0.2222 & -0.3333 \\ -0.2222 & 0.1111 & -0.6667 \\ -0.1852 & -0.0741 & -0.2222 \end{bmatrix}$$ ### Solution to the System $$Ax = b$$ By multiplying the inverse $$A^{-1}$$ with vector $$b$$, we get: $$x = A^{-1}b = \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix}$$ ### Conclusion The solution to the system $$Ax = b$$ is: $$x = \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix}$$ Would you like more details on any of these steps, or do you have any questions? ### Related Questions: 1. How do you compute the determinant of a matrix? 2. What is the significance of a matrix being invertible? 3. How can we find the adjugate of a matrix? 4. What methods can be used to solve systems of linear equations besides matrix inversion? 5. How do you perform matrix multiplication? ### Tip: When dealing with larger matrices, numerical methods like LU decomposition or using software tools are more efficient for finding inverses and solving linear systems.