Solve each system using matrices. If there is no solution or...

Solve each system using matrices. If there is no solution or if there are infinitely many solutions and a system’s equations are dependent, so state.
$\left\{\begin{aligned} x-3 y+z &=2 \\ 4 x-12 y+4 z &=8 \\-2 x+6 y-2 z &=-4 \end{aligned}\right.$

Key Concepts

Matrix Representation

Matrix representation is the process of writing a system of linear equations in a compact form using matrices. This involves aligning the coefficients of the variables into a structured array (the coefficient matrix), and optionally augmenting it with the constants from the equations to form an augmented matrix. This representation is foundational for applying matrix techniques to solve linear systems efficiently.

Augmented Matrix

An augmented matrix is a matrix that includes both the coefficients of the variables and the constants from the system of equations. It provides a single framework for performing operations on the entire system, facilitating the use of algorithmic procedures such as Gaussian elimination to simplify the system and determine its solutions.

Gaussian Elimination

Gaussian elimination is a systematic method used to simplify a system of linear equations by performing elementary row operations on the augmented matrix. These operations—row swapping, scaling, and row addition—aim to convert the matrix into an upper triangular (or row echelon) form. This form makes it easier to solve for the variables systematically either through back substitution or further reduction into reduced row echelon form.

Elementary Row Operations

Elementary row operations are the basic manipulations allowed on the rows of a matrix that do not change the solution set of the corresponding system of equations. These operations include interchanging two rows, multiplying a row by a nonzero number, and adding or subtracting a multiple of one row to another. They are essential for methods like Gaussian elimination because they enable the systematic simplification of the matrix.

Rank of a Matrix and Consistency

The rank of a matrix, which is the number of linearly independent rows (or columns) in the matrix, is crucial in determining the consistency of a system of linear equations. Comparing the rank of the coefficient matrix with the rank of the augmented matrix helps establish whether the system has a unique solution, infinitely many solutions, or no solution. This concept provides a theoretical basis for the results obtained through matrix operations.

Dependent Equations and Infinitely Many Solutions

Dependent equations occur when one or more equations in the system can be derived from a combination of the other equations. This leads to redundancy in the system, which typically results in infinitely many solutions. Recognizing dependency is important because it informs how the degrees of freedom will affect the final solution set and helps determine whether the system is underdetermined (having infinitely many solutions) or inconsistent.

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