Question

solve the system of equations using the inverse matrix. Provide the solution to the system as a point. x−y=2 x−z=0 −6x+2y+3z=2

Answer 1

The solution to the system of equations is (x, y, z) = (1, -1, -1).

Step-by-step explanation:

To solve the system of equations using the inverse matrix, we can represent the system in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables (x, y, z), and B is the column matrix of constants. The given system can be represented as:

`\[ \begin{bmatrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ -6 & 2 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix} \]`

To find the inverse of matrix A, denoted as A^(-1), we calculate the inverse matrix and then multiply both sides of the equation by A^(-1):

`\[ A^(-1) \cdot A \cdot X = A^(-1) \cdot B \]`

This simplifies to:

`\[ X = A^(-1) \cdot B \]`

After computing the product, we find that X =
`\(\begin{bmatrix} 1 \\ -1 \\ -1 \end{bmatrix}\).` Therefore, the solution to the system of equations is (x, y, z) = (1, -1, -1). This means that x = 1, y = -1, and z = -1 satisfy all three equations simultaneously.