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.x - y + z = 82y - z = - 72x + 3y = 1The inverse of is

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1. Write the system of equations in matrix form: AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants.
2. Identify the coefficient matrix A as \( \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \end{bmatrix} \), the variable matrix X as \( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \), and the constant matrix B as \( \begin{bmatrix} 8 \\ -7 \\ 1 \end{bmatrix} \).
3. Express the matrix equation as \( \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 8 \\ -7 \\ 1 \end{bmatrix} \).
4. Use the given inverse matrix \( \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \) to solve for X by multiplying both sides of the equation by the inverse of A.
5. Calculate \( X = A^{-1}B \) by performing the matrix multiplication \( \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \begin{bmatrix} 8 \\ -7 \\ 1 \end{bmatrix} \) to find the values of x, y, and z.

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 is invertible, the solution can be found using X = A⁻¹B. The existence of an inverse is crucial for solving linear systems using this method.

Solving Linear Systems

Solving linear systems involves finding the values of the variables that satisfy all equations simultaneously. Techniques include substitution, elimination, and using matrix methods such as finding the inverse of the coefficient matrix. Understanding these methods is essential for effectively addressing problems in college algebra.