A. Write each linear system as a matrix equation in the form...

A. Write each linear system as a matrix equation in the form $A X=B$
B. Solve the system using the inverse that is given for the coefficient matrix.
$$
\left\{\begin{array}{l}
{2 x+6 y+6 z=8} \\
{2 x+7 y+6 z=10} \\
{2 x+7 y+7 z=9}
\end{array}\right.
$$

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Key Concepts

Inverse Matrix Method

When the coefficient matrix A is invertible, the system of linear equations can be solved by computing its inverse, A?¹, and then obtaining the solution vector X by the product A?¹·B. This method leverages the property that A·A?¹ equals the identity matrix, thereby isolating the vector X, and it is a powerful technique in solving linear systems, provided that the inverse exists.

Coefficient Matrix and Invertibility

The coefficient matrix plays a central role in determining the characteristics of the system, such as whether a unique solution exists. Invertibility of the coefficient matrix, often verified by a non-zero determinant, is crucial because it confirms that the inverse exists and that the system has one unique solution. This concept is key in understanding why and when the inverse matrix method can be applied.

Matrix Equation Representation

This concept involves expressing a system of linear equations in the form A·X = B, where A is the coefficient matrix composed of the coefficients of the variables, X is the vector of variables, and B is the constant vector. Representing a system this way streamlines the process of applying linear algebra techniques and makes it possible to use matrix operations to solve the system.

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