To solve the system of equations using the inverse matrix method, we follow these steps:

We are given the following system of linear equations:

$5x - 6y + 4z = 15$ $7x + 4y - 3z = 19$ $2x + y + 6z = 46$

Step 1: Write the system in matrix form.

This system can be written as:

$\begin{bmatrix} 5 & -6 & 4 \\ 7 & 4 & -3 \\ 2 & 1 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 15 \\ 19 \\ 46 \end{bmatrix}$

Here, the matrix $A$ is:

$A = \begin{bmatrix} 5 & -6 & 4 \\ 7 & 4 & -3 \\ 2 & 1 & 6 \end{bmatrix}$

The vector of variables $\mathbf{x}$ is:

$\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$

And the constant vector $\mathbf{b}$ is:

$\mathbf{b} = \begin{bmatrix} 15 \\ 19 \\ 46 \end{bmatrix}$

Step 2: Find the inverse of matrix $A$.

To solve for $\mathbf{x}$, we use the formula:

$\mathbf{x} = A^{-1} \mathbf{b}$

Step 3: Calculate the inverse of matrix $A$.

Let's find the inverse of matrix $A$ and then multiply it with vector $\mathbf{b}$ to get the values of $x$, $y$, and $z$.

I'll proceed with the calculations now. numpy as np

Define the matrix A and vector b

A = np.array([[5, -6, 4], [7, 4, -3], [2, 1, 6]]) b = np.array([15, 19, 46])

Calculate the inverse of A

A_inv = np.linalg.inv(A)

Solve for x, y, z

x_y_z = np.dot(A_inv, b) x_y_zThe solution to the system of equations is:

$x = 3, \quad y = 4, \quad z = 6$

Would you like further clarification on any of the steps or details about the calculations?

Here are some related questions you might find helpful:

1. How do you compute the inverse of a matrix using elementary row operations?
2. What does it mean for a matrix to be invertible, and how can we determine that?
3. Can this method be applied to non-square systems of equations?
4. How do you use matrix multiplication to solve a system of linear equations?
5. How can you verify the solution to a system of linear equations?

Tip: When solving systems using the inverse matrix method, ensure that the matrix is invertible, i.e., its determinant is non-zero, or else the system might have no unique solution.