The given matrix equation is:

$A \cdot X = B$

where

$A = \begin{bmatrix} -1 & 2 \\ -1 & 4 \\ -2 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 6 \\ 8 \\ 2 \end{bmatrix}$

To solve for $X$, we need to check if $A$ is invertible. If it is, we can find $X$ using:

$X = A^{-1} B$

Let's compute this. numpy as np

Define matrices A and B

A = np.array([[-1, 2], [-1, 4], [-2, 0]]) B = np.array([[6], [8], [2]])

Check if A is square and invertible

if A.shape[0] == A.shape[1]: det_A = np.linalg.det(A) else: det_A = None

det_AMatrix $A$ is not a square matrix (it has dimensions $3 \times 2$), so it is not invertible. This means we need to solve the system using a least-squares approach or check if it has infinitely many solutions. Let's proceed with finding the least-squares solution. Compute the least-squares solution (X) using the pseudo-inverse of A X, residuals, rank, s = np.linalg.lstsq(A, B, rcond=None) XThe least-squares solution for $X$ is:

$X = \begin{bmatrix} -1.14 \\ 1.86 \end{bmatrix}$

(Rounded to two decimal places)

Would you like a detailed step-by-step explanation?

Related Questions:

1. What does it mean for a system to have no exact solution but a least-squares solution?
2. How is the pseudo-inverse of a matrix used in solving non-square systems?
3. What are the conditions for a matrix to be invertible?
4. How can you check if a system has infinitely many solutions?
5. How can singular value decomposition (SVD) help in solving linear systems?

Tip: When a system of equations is overdetermined ($m > n$), a least-squares solution minimizes the sum of squared residuals.