Consider the following system of linear equations:
2x1+2x2+6x3	 = 	−14
−x1+4x2+2x3	 = 	−13
3x1+7x3	 = 	−10

Let A be the coefficient matrix and X the solution matrix to the system. Solve the system by first computing A−1 and then using it to find X.

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To solve the given system of linear equations using matrix inversion, let's proceed with the following steps:

### Step 1: Write the System in Matrix Form
The given system is:

\[
2x_1 + 2x_2 + 6x_3 = -14
\]
\[
-x_1 + 4x_2 + 2x_3 = -13
\]
\[
3x_1 + 0x_2 + 7x_3 = -10
\]

We can write this system in matrix form as \( A \cdot X = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix.

\[
A = \begin{bmatrix} 2 & 2 & 6 \\ -1 & 4 & 2 \\ 3 & 0 & 7 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad B = \begin{bmatrix} -14 \\ -13 \\ -10 \end{bmatrix}
\]

### Step 2: Find the Inverse of Matrix A (\( A^{-1} \))
To solve for \( X \), we need to compute the inverse of \( A \), if it exists.

Let's compute \( A^{-1} \).

The inverse of matrix \( A \) is:

\[
A^{-1} = \begin{bmatrix} 2.8 & -1.4 & -2.0 \\ 1.3 & -0.4 & -1.0 \\ -1.2 & 0.6 & 1.0 \end{bmatrix}
\]

### Step 3: Solve for Matrix \( X \)
To find the solution matrix \( X \), we use the equation:

\[
X = A^{-1} \cdot B
\]

Let's calculate \( X \).The solution matrix \( X \) is:

\[
X = \begin{bmatrix} -1 \\ -3 \\ -1 \end{bmatrix}
\]

Thus, the solutions are:
\[
x_1 = -1, \quad x_2 = -3, \quad x_3 = -1
\]

Would you like more details or have any questions about this process?

### Related Questions:
1. How can we determine if a matrix is invertible before attempting to find its inverse?
2. What is the significance of using matrix inversion to solve a system of linear equations?
3. Can you solve the system using other methods, such as Gaussian elimination?
4. What happens if the determinant of the matrix \( A \) is zero?
5. How can we interpret the solution geometrically if there are infinitely many solutions?

**Tip:** Always verify if a matrix is invertible by checking if its determinant is non-zero before attempting to compute its inverse.

Consider the following system of linear equations: 2x1+2x2+6x3 = −14, −x1+4x2+2x3 = −13, 3x1+7x3 = −10. Let A be the coefficient matrix and X the solution matrix to the system. Solve the system by first computing A−1 and then using it to find X.

Learn how to solve a system of linear equations by computing the inverse of a matrix and using it to find the solution. This problem covers key linear algebra concepts, including matrix inversion and properties of systems of equations. Suitable for advanced high school or undergraduate students.

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