In Exercises 37 - 42,a. Write each linear system as a matrix equation in the form AX = B.b. Solve the system using the inverse that is given for the coefficient matrix.w - x + 2y = - 3x - y + z = 4- w + x - y + 2z = 2- x + y - 2z = - 4The inverse of is

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1. Identify the coefficient matrix A from the system of equations: A = \begin{bmatrix} 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \end{bmatrix}.
2. Identify the variable matrix X: X = \begin{bmatrix} w \\ x \\ y \\ z \end{bmatrix}.
3. Identify the constant matrix B: B = \begin{bmatrix} -3 \\ 4 \\ 2 \\ -4 \end{bmatrix}.
4. Write the matrix equation in the form AX = B.
5. Use the given inverse of matrix A to solve for X by calculating X = A^{-1}B.

Key Concepts

Matrix Equation

A matrix equation is a mathematical representation of a system of linear equations in the form AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants. This format allows for efficient manipulation and solution of the system using matrix operations.

Inverse of a Matrix

The inverse of a matrix A, denoted as A⁻¹, is a matrix that, when multiplied by A, yields the identity matrix. For a system of equations represented as AX = B, if A has an inverse, the solution can be found using X = A⁻¹B. The existence of an inverse is contingent on A being a square matrix with a non-zero determinant.

Solving Linear Systems

Solving linear systems involves finding the values of the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and using matrix operations. In this case, the inverse matrix method is applied, which simplifies the process by transforming the system into a single equation that can be easily solved.