Find the inverse of the Matrix  A=[-2 2 -3 2 1 -6 1 -2 0]and hence solve the system Ax=b where b=[1 2 3]T

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:

\[
A^{-1} = \begin{bmatrix} 
-0.4444 & 0.2222 & -0.3333 \\ 
-0.2222 & 0.1111 & -0.6667 \\ 
-0.1852 & -0.0741 & -0.2222 
\end{bmatrix}
\]

### 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.

Find the inverse of the Matrix A=[-2 2 -3 2 1 -6 1 -2 0] and hence solve the system Ax=b where b=[1 2 3]^T

Discover how to find the inverse of a 3x3 matrix A and solve the system of linear equations Ax=b. This problem delves into linear algebra concepts, including matrix inversion, determinants, and adjugates. Suitable for students in Grades 10-12.

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