Problem

Statement

Study the following system of linear equations dependent on the real parameter $( a )$ and solve it in the cases where it is compatible:

$$ \begin{cases} (a - 1)x - y = 3 \ (a - 1)x + (a + 1)y - (2 - a)z = -2a \ (-2a + 2)x - ay + (a^2 - a - 2)z = 3a - 1 \end{cases} $$

Mention the theoretical result employed and justify its use.

Solution

To solve the given system of linear equations dependent on the parameter $(a)$:

$$ \begin{cases} (a - 1)x - y = 3 \ (a - 1)x + (a + 1)y - (2 - a)z = -2a \ (-2a + 2)x - ay + (a^2 - a - 2)z = 3a - 1 \end{cases} $$

we will analyze the system for different values of $(a)$ to determine when the system is compatible (i.e., has a solution).

Step 1: Write the augmented matrix

The system of equations can be written in matrix form as $(A \mathbf{x} = \mathbf{b})$:

A = \begin{pmatrix}
  a - 1 &     -1 &           0 \\
  a - 1 &  a + 1 &       a - 2 \\
-2a + 2 & -a     & a^2 - a - 2
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix} x, \\ y, \\ z \end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix} 3, \\ -2a, \\ 3a - 1 \end{pmatrix}

The augmented matrix $([A|\mathbf{b}])$ is:

\left(\begin{array}{ccc|c}  
  a - 1 &     -1 &           0   &       3 \\
  a - 1 &  a + 1 &       a - 2   & -2a       \\
-2a + 2 & -a     & a^2 - a - 2   &  3a - 1
\end{array}\right)

Step 2: Perform row operations to simplify the augmented matrix

1. Subtract row 1 from row 2:

R2 \to R2 - R1 = \begin{pmatrix} 

0 && a + 2 && -(2 - a) && -2a - 3 

\end{pmatrix}

2. Add 2 times row 1 to row 3:

R3 \to R3 + 2R1 = \begin{pmatrix}

 0 && -a - 2 && a^2 - a - 2 && 3a + 5 

\end{pmatrix}

The augmented matrix is now:

\left(\begin{array}{ccc|c}
a - 1 & -1     & 0            &       3 \\
    0 &  a + 2 &       a - 2  & -2a - 3 \\
    0 & -a - 2 & a^2 - a - 2  &  3a + 5
\end{array}\right)

3. Add row 2 to row 3:

R3 \to R3 + R2 = \begin{pmatrix} 

0  &  0 &  a^2 - 4  &  a + 2 

\end{pmatrix}

The augmented matrix is now:

\left(\begin{array}{ccc|c}

a - 1 &   - 1 &           0 &      3 \\
    0 & a + 2 &      a - 2 & -2a - 3 \\
    0 &     0 & a^2    - 4 &   a + 2

\end{array}\right)

Step 3: Analyze the resulting matrix

For the system to be compatible, we need the last row to not contradict (i.e., it should not be of the form $([0 ; 0 ; 0 ; | ; c])$ with $(c \neq 0))$.

1. Check the coefficient of $(z)$ in the last row: $(a^2 - a - 4)$:

a^2 - 4 = (a - 2)(a + 2)

• For $(a = 2)$, the coefficient is zero.
• For $(a = -2)$, the coefficient is zero.

Let's consider these cases:

Case 1: $(a = 2)$

For $(a = 2)$, the system becomes:

\begin{cases}
 1x  - y   =  3 \\
 1x + 3y  =  -4 \\
-2x - 2y  =   5
\end{cases}

The augmented matrix for $(a = 2)$ is:

\left(\begin{array}{ccc|c}
 1 & -1 & 0 &  3 \\
 1 &  3 & 0 & -4 \\
-2 & -2 & 0 &  5
\end{array}\right)

Perform row operations:

1. Subtract row 1 from row 2:

R2 \to R2 - R1 = \begin{pmatrix} 

0 & 4 & 0 & -7 

\end{pmatrix}

2. Add 2 times row 1 to row 3:

R3 \to R3 + 2R1 = \begin{pmatrix} 

0 & -4 & 0 & 11 

\end{pmatrix}

The matrix is:

\left(\begin{array}{ccc|c}

1 & -1 & 0 &  3 \\
0 &  4 & 0 & -7 \\
0 & -4 & 0 & 11

\end{array}\right)

Since the last two rows are contradictory $((4y = -7)$ and $(-4y = 11))$, the system is incompatible for $(a = 2)$.

Case 2: $(a = -2)$

For $(a = -2)$, the system becomes:

\begin{cases}
-3x  - y      =  3 \\
-3x  + y      =  4 \\
 6x  + 2y     = -7
\end{cases}

\left(\begin{array}{ccc|c}
-3 & -1 & 0 & 3 \\
-3 &  1 & 0 & 4 \\
 6 &  2 & 0 & -7
\end{array}\right)

Perform row operations:

1. Subtract row 1 from row 2:

R2 \to R2 - R1 = \begin{pmatrix} 

0 & 2 & 0 & 1 

\end{pmatrix}

2. Add 2 times row 1 to row 3:

R3 \to R3 + 2R1 = \begin{pmatrix} 

0 & 0 & 0 & -1 

\end{pmatrix}

Since the last row is contradictory ( $(0 = -1)$ ), the system is incompatible for $(a = -2)$.

General Case: $(a \neq 2) and (a \neq -2)$

For other values of $(a)$, the coefficient $(a^2 - 4 \neq 0)$, so the system can be solved by substitution or other methods. Since the equations are not contradictory, we can find the solution by solving the resulting triangular matrix.

Hence, the system is compatible for all $(a \neq 2)$ and $(a \neq -2)$.

Theoretical Justification

The method used involves analyzing the rank of the coefficient matrix and the augmented matrix to determine compatibility. This is justified by the Rouché–Capelli theorem, which states that a system of linear equations is compatible if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix.

We systematically performed row operations to transform the augmented matrix into row-echelon form and determined the values of (a) that lead to compatible or incompatible systems based on the resulting form.

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