Question

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{r} x+2 y-z=3 \\ 2 x-y+2 z=6 \\ x-3 y+3 z=4 \end{array}\right. $$

Short answer

The system is inconsistent and has no solution.

Step-by-step solution

Write the augmented matrix

Transform the system of equations into an augmented matrix. The system is given by: o (1) \ x + 2y - z = 3 \ (2) \ 2x - y + 2z = 6 \ (3) \ x - 3y + 3z = 4. The augmented matrix is: o \[ \left\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 2 & -1 & 2 & 6 \ 1 & -3 & 3 & 4 \end{array} \right\]. \]

Apply row operations to obtain the row echelon form

Use row operations to simplify the augmented matrix. Start by making the element in \text{row 2, column 1} to zero. Subtract 2 times the first row from the second row:o \[ R2 \rightarrow R2 - 2R1. \]The matrix becomes: o \[ \left\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 0 & -5 & 4 & 0 \ 1 & -3 & 3 & 4 \end{array} \right\]. \]

Eliminate the element in row 3, column 1

Subtract the first row from the third row:o \[ R3 \rightarrow R3 - R1. \]The matrix becomes: o \[ \left\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 0 & -5 & 4 & 0 \ 0 & -5 & 4 & 1 \end{array} \right\]. \]

Eliminate the element in row 3, column 2

Subtract the second row from the third row:o \[ R3 \rightarrow R3 - R2. \]The matrix becomes: o \[ \left\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 0 & -5 & 4 & 0 \ 0 & 0 & 0 & 1 \end{array} \right\]. \]

Determine consistency

The last row of the matrix corresponds to the equation o \[ 0 = 1 \]which is a contradiction. This indicates that the system is inconsistent and has no solution.

Key concepts

Row Operations

Row operations are techniques used to manipulate the rows of a matrix to facilitate solving systems of linear equations. They include three main types:

• Row swapping: Exchanging two rows
• Row multiplication: Multiplying all elements of a row by a nonzero scalar
• Row addition or subtraction: Adding or subtracting a multiple of one row to another row

By strategically applying these operations, we can simplify the matrix to a form that reveals the solutions to the system.
In our exercise, we used row operations to eliminate specific elements to eventually find contradictions, which indicate that the system is inconsistent.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations. It includes coefficients of the variables from the left side of each equation, as well as the constants on the right side.
For example, the system of equations:
\( \begin{cases} x + 2y - z = 3 & \ 2x - y + 2z = 6 & \ x - 3y + 3z = 4 \ end{cases} \)
can be written as the augmented matrix:
\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 2 & -1 & 2 & 6 \ 1 & -3 & 3 & 4 \end{array} \]
This helps to easily apply row operations to solve the system.

Inconsistent System

An inconsistent system of equations is one that has no solution. This occurs when the equations contradict each other, meaning there is no point that satisfies all the equations simultaneously.
In our example, after transforming the augmented matrix through row operations, we arrived at the final form:
\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 0 & -5 & 4 & 0 \ 0 & 0 & 0 & 1 \end{array} \]
The last row corresponds to the equation \0 = 1, which is clearly a contradiction. This indicates that the system has no possible solutions and is, therefore, inconsistent.

Row Echelon Form

The row echelon form (REF) of a matrix is a type of matrix that is easier to work with for solving systems of equations. It has the following rules:

• All rows consisting entirely of zeros are at the bottom.
• In each non-zero row, the leading entry (first non-zero number) is to the right of the leading entry of the row above it.
• The leading entry in any non-zero row is 1, known as the pivot.
• The leading entry is the only non-zero entry in its column.

Converting a matrix to REF can greatly simplify solving the system of equations.
In our exercise, through a series of row operations, we aimed to reach such a form to determine the solutions or identify inconsistency.