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.