Problem 50 - Discrete Mathematics with Applications

Question

Rewrite each linear system as a matrix equation \(A X=B\). $$\begin{aligned} x-2 y &=4 \\ 3 x+y-z &=-5 \\ x+2 y-3 z &=6 \end{aligned}$$

Step-by-Step Solution

Answer

The given linear system can be rewritten as a matrix equation \(AX = B\) as follows: \[ \begin{bmatrix} 1 & -2 & 0\\ 3 & 1 & -1\\ 1 & 2 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4\\ -5\\ 6 \end{bmatrix} \]

Step 1: Identify the coefficients and constants in the given linear system

From the given linear system, we can identify the coefficients of the variables and the constants as follows: \[\begin{aligned} x - 2y &= 4 \ \ (1) \\ 3x + y - z &= -5 \ \ (2) \\ x + 2y - 3z &= 6 \ \ (3) \end{aligned}\]

Step 2: Form the coefficient matrix \(A\), variable matrix \(X\), and constant matrix \(B\)

From the given linear system, we can form the coefficient matrix \(A\) by arranging the coefficients of the variables x, y, and z in the equations. Then, form the variable matrix \(X\) by arranging the variables x, y, and z, and finally form the constant matrix \(B\) by arranging the constants from the equations. \[\begin{aligned} A &= \begin{bmatrix} 1 & -2 & 0\\ 3 & 1 & -1\\ 1 & 2 & -3 \end{bmatrix}, \qquad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \qquad B = \begin{bmatrix} 4\\ -5\\ 6 \end{bmatrix} \end{aligned}\]

Step 3: Write the matrix equation \(AX = B\)

Now that we have the matrices \(A\), \(X\), and \(B\), we can write the matrix equation as follows: \[AX = B\]

Step 4: Substitute the matrices and write the final matrix equation

Finally, substitute the values of the matrices \(A\), \(X\), and \(B\) in the matrix equation to get the final matrix equation. \[\begin{bmatrix} 1 & -2 & 0\\ 3 & 1 & -1\\ 1 & 2 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4\\ -5\\ 6 \end{bmatrix}\]

Key Concepts

Linear Algebra

Linear algebra is a fundamental branch of mathematics that focuses on vectors, vector spaces, linear transformations, and systems of linear equations. It is a key concept not only in mathematics but also in fields such as physics, computer science, and engineering. It allows us to translate real-world problems into mathematical language, often using matrices and vectors to simplify and solve complex problems.

Coefficient Matrix

A coefficient matrix is an arrangement of coefficients that appear in a system of linear equations. In the case of the exercise example, the coefficient matrix, denoted as \(A\), encapsulates all the multipliers for the variables \(x\), \(y\), and \(z\) in a compact, rectangular array. It's a convenient way to represent the system using a neat, tabular format.

By listing out coefficients in this way, we fundamentally transform the problem into a format that's well-suited for various matrix operations, such as multiplication, inversion, or applying row operations. These techniques form the backbone of many linear algebra methods used to solve systems of equations.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find a common solution, which means assigning values to the variables to satisfy all equations simultaneously.

Systems like this can graphically represent lines, planes, or hyperplanes in space, where their intersection corresponds to the solution set. In our exercise, the matrix equation \(AX = B\) is a compact way to represent such a system, where \(X\) is the set of variables, \(A\) is a matrix containing all coefficients, and \(B\) is a column matrix representing the constants. Solving matrix equations often involve methods like substitution, elimination, or using more advanced techniques such as matrix inversion or applying Gaussian elimination.