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. $$\left\{\begin{aligned}x+2 y+5 z &=2 \\2 x+3 y+8 z &=3 \\-x+y+2 z &=3 \end{aligned}\right.$$ The inverse of $\left[\begin{array}{rrr}1 & 2 & 5 \\ 2 & 3 & 8 \\ -1 & 1 & 2\end{array}\right]$ is $\left[\begin{array}{rrr}2 & -1 & -1 \\ 12 & -7 & -2 \\ -5 & 3 & 1\end{array}\right]$

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.
$$\left\{\begin{aligned}x+2 y+5 z &=2 \\2 x+3 y+8 z &=3 \\-x+y+2 z &=3 \end{aligned}\right.$$
The inverse of
$\left[\begin{array}{rrr}1 & 2 & 5 \\ 2 & 3 & 8 \\ -1 & 1 & 2\end{array}\right]$ is $\left[\begin{array}{rrr}2 & -1 & -1 \\ 12 & -7 & -2 \\ -5 & 3 & 1\end{array}\right]$

Key Concepts

Matrix Equation

A matrix equation is a compact representation of a system of linear equations, typically expressed in the form AX = B, where A is the coefficient matrix containing the coefficients of the variables, X is the column vector of unknowns, and B is the column vector of constants. This representation facilitates the use of linear algebraic techniques, including matrix operations, to analyze and solve the system.

Coefficient Matrix

The coefficient matrix is the matrix formed by the coefficients of the variables in a system of linear equations. It is a fundamental component in representing the system as AX = B. The properties of the coefficient matrix, such as its determinant and invertibility, play a crucial role in determining whether the system has a unique solution, no solution, or infinitely many solutions.

Matrix Inverse

The matrix inverse of a square matrix A is another matrix, denoted A?¹, such that when it is multiplied by A, it yields the identity matrix. In the context of solving a system of linear equations, if the coefficient matrix A is invertible, the inverse can be used to directly find the solution vector X by the computation X = A?¹B.

Solving Linear Systems using Matrix Inverse

When the coefficient matrix of a system of linear equations is invertible, the system can be solved by multiplying both sides of the equation AX = B by the inverse of A. This method results in the solution X = A?¹B. It is an efficient and systematic approach that leverages matrix multiplication and the concept of the inverse to find the values of the variables in the system.

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