Write each system in the form A X=B . Then solve the system...

Write each system in the form $A X=B .$ Then solve the system by entering $A$ and $B$ into your graphing utility and computing $A^{-1} B$
$$
\left\{\begin{array}{r}
{w+x+y+z=4} \\
{w+3 x-2 y+2 z=7} \\
{2 w+2 x+y+z=3} \\
{w-x+2 y+3 z=5}
\end{array}\right.
$$

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Key Concepts

Coefficient Matrix

The coefficient matrix (A) is a rectangular or square matrix that contains the coefficients of the variables in the system of linear equations. Its structure plays a crucial role in the properties of the system, including whether it has a unique solution, infinitely many solutions, or no solution.

Matrix Inverse Method

The matrix inverse method is used to solve systems of linear equations when the coefficient matrix A is invertible. The unique solution is given by X = A?¹B, where A?¹ is the inverse of the matrix A. This method is particularly useful for systems where A is a square matrix and its inverse exists.

System of Linear Equations

A system of linear equations consists of multiple equations with several variables where each equation expresses a linear relationship between the variables. It is fundamental in linear algebra and appears in diverse applications such as engineering, economics, and physics.

Matrix Representation

This concept involves writing the system of equations in the compact form AX = B, where A is the coefficient matrix containing the coefficients of the variables, X is the column vector of variables, and B is the constant vector. This representation simplifies manipulation and solution of the system using matrix operations.

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