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. $$\left\{\begin{aligned}w-x+2 y &=-3 \\x-y+z &=4 \\-w+x-y+2 z &=2 \\ -x+y-2 z &=-4\end{aligned}\right.$$ The inverse of $\left[\begin{array}{rrrr}1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2\end{array}\right]$ is $\left[\begin{array}{rrrr}0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1\end{array}\right]$

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.
$$\left\{\begin{aligned}w-x+2 y &=-3 \\x-y+z &=4 \\-w+x-y+2 z &=2 \\
-x+y-2 z &=-4\end{aligned}\right.$$
The inverse of
$\left[\begin{array}{rrrr}1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2\end{array}\right]$ is $\left[\begin{array}{rrrr}0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1\end{array}\right]$

Key Concepts

Matrix Equation

A matrix equation is a compact representation of a system of linear equations where the coefficients of the variables are arranged in a matrix (often called the coefficient matrix), the variables are contained in a column vector, and the constants form another column vector. It is typically written in the form A X = B, which simplifies many operations in linear algebra.

Coefficient Matrix

The coefficient matrix is the matrix constructed from the coefficients of the variables in a system of linear equations. It encapsulates the structure of the system and is the key component in many methods for obtaining solutions, such as those using matrix inverses or row reduction.

Matrix Inverse

The matrix inverse is a matrix which, when multiplied with the original square matrix, results in the identity matrix. This property is crucial for solving matrix equations, as it allows one to 'cancel out' the coefficient matrix by multiplying both sides of the equation by its inverse, provided the matrix is non-singular.

Solving Linear Systems Using the Inverse

When the coefficient matrix in a system of linear equations is invertible, the system can be solved by multiplying both sides of the matrix equation A X = B by the inverse of A. The solution is then given directly as X = A?¹B, which provides a straightforward method to obtain the variables' values.

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