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. $$\left\{\begin{aligned}x-6 y+3 z &=11 \\2 x-7 y+3 z &=14 \\4 x-12 y+5 z &=25 \end{aligned}\right.$$ The inverse of $\left[\begin{array}{rrr}1 & -6 & 3 \\ 2 & -7 & 3 \\ 4 & -12 & 5\end{array}\right]$ is $\left[\begin{array}{rrr}1 & -6 & 3 \\ 2 & -7 & 3 \\ 4 & -12 & 5\end{array}\right]$

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.
$$\left\{\begin{aligned}x-6 y+3 z &=11 \\2 x-7 y+3 z &=14 \\4 x-12 y+5 z &=25 \end{aligned}\right.$$
The inverse of $\left[\begin{array}{rrr}1 & -6 & 3 \\ 2 & -7 & 3 \\ 4 & -12 & 5\end{array}\right]$ is $\left[\begin{array}{rrr}1 & -6 & 3 \\ 2 & -7 & 3 \\ 4 & -12 & 5\end{array}\right]$

Key Concepts

Solving Linear Systems Using Matrix Inversion

When a system of linear equations can be arranged as A·X = B and the coefficient matrix A is invertible, the unique solution is obtained by directly multiplying the inverse of A by B. This method offers a systematic approach to finding the solution and is particularly effective for computer-based implementations where matrix operations are optimized.

Matrix Inverse

The inverse of a square matrix is another matrix that, when multiplied with the original matrix, yields the identity matrix. In the context of solving linear systems, if the coefficient matrix A is invertible, then the system A·X = B can be solved by computing X = A?¹·B. Finding the inverse is a key step in many analytic solutions in linear algebra.

Coefficient Matrix

The coefficient matrix is the matrix that collects all the coefficients of the variables from each equation in the system. Arranging the coefficients in matrix form allows one to analyze and solve the system with techniques from matrix algebra, and is crucial when applying operations like finding the inverse or performing elimination procedures.

Matrix Equation

A matrix equation represents a system of linear equations in a compact form, typically written as A·X = B, where A is the coefficient matrix containing the coefficients of the variables, X is the column vector of the variables, and B is the column vector representing the constants from each equation. This formulation is fundamental in linear algebra since it enables efficient manipulation and solution of systems using matrix operations.

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