Consider the system of equations
x-2y+3z=-1
-x+y-2z=k
x-3y+4z=1
Statement -1 The system of equation has no solutions for `k ne 3`.
statement -2 The determinant `|{:(1,3,-1),(-1,-2,k),(1,4,1):}| ne0, "for"" " kne3.`

Options

A. (a)Satement -1 is true Statement -2 is true and Statement -2 is correct explanation for Statement -1.
B. (b)Satement -1 is true Statement -2 is true and Statement -2 is not a correct explanation for Statement -1.
C. (c)Statement -1 is true Statement -2 is false.
D. (d)Statement-1 is false, Statement -2 is true.

Answer

To solve the given system of equations and analyze the statements, we will follow these steps:

### Step 1: Write the system of equations in matrix form
The given equations are:
1. \( x - 2y + 3z = -1 \)
2. \( -x + y - 2z = k \)
3. \( x - 3y + 4z = 1 \)

We can represent this system in the form of a matrix equation \( A \mathbf{x} = \mathbf{b} \), where:
\[ A = \begin{pmatrix} 1 & -2 & 3 \\ -1 & 1 & -2 \\ 1 & -3 & 4 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -1 \\ k \\ 1 \end{pmatrix} \]

### Step 2: Calculate the determinant of matrix \( A \)
To determine if the system has a unique solution, we need to calculate the determinant of matrix \( A \):
\[ |A| = \begin{vmatrix} 1 & -2 & 3 \\ -1 & 1 & -2 \\ 1 & -3 & 4 \end{vmatrix} \]

Using the determinant formula for a 3x3 matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the elements of the second and third rows respectively.

Calculating:
\[ |A| = 1 \cdot (1 \cdot 4 - (-2) \cdot (-3)) - (-2) \cdot (-1 \cdot 4 - (-2) \cdot 1) + 3 \cdot (-1 \cdot (-3) - 1 \cdot 1) \]
\[ = 1 \cdot (4 - 6) + 2 \cdot (4 - 2) + 3 \cdot (3 - 1) \]
\[ = 1 \cdot (-2) + 2 \cdot 2 + 3 \cdot 2 \]
\[ = -2 + 4 + 6 = 8 \]

### Step 3: Analyze the conditions for no solutions
For the system to have no solutions, the determinant of the augmented matrix (which includes the constants from the right side of the equations) must equal zero while the determinant of the coefficient matrix is non-zero.

The augmented matrix is:
\[ \begin{pmatrix} 1 & -2 & 3 & | & -1 \\ -1 & 1 & -2 & | & k \\ 1 & -3 & 4 & | & 1 \end{pmatrix} \]

### Step 4: Set up the determinant of the augmented matrix
The determinant of the augmented matrix can be calculated similarly, but we can also derive conditions directly from the determinant of the coefficient matrix.

### Step 5: Conclusion about the statements
1. **Statement 1**: "The system of equation has no solutions for \( k \neq 3 \)" is true because we found that the determinant of the coefficient matrix is non-zero, and the system can only be inconsistent for specific values of \( k \).

2. **Statement 2**: "The determinant \( |A| \neq 0 \) for \( k \neq 3 \)" is true as well, since we have shown that the determinant is non-zero except for the specific case when \( k = 3 \).

### Final Answer
Both statements are true, and statement 2 correctly explains statement 1.

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