Let S be the set of all real values of k for which the system of linear equations x+y+z=2, 2x+y–z=3,

Q: Wait, why do we only care about the determinant of the coefficient matrix? What about the constants on the right side?

That is a fantastic question! For a system of linear equations $AX = B$, the condition for a unique solution depends solely on the coefficient matrix $A$. If $\Delta \neq 0$, the matrix $A$ is invertible, meaning $X = A^{-1}B$ has a unique solution regardless of the constants in $B$. The constants only matter when $\Delta = 0$, where they determine if the system has no solution or infinitely many solutions.

Q: What if the determinant is zero? Does that mean there are no solutions?

Not necessarily! If $\Delta = 0$, the system is either inconsistent (no solution) or dependent (infinitely many solutions). To distinguish between these, you would need to check the consistency using the augmented matrix or by substituting the value of $k$ back into the equations. In this specific problem, we only need to find the values of $k$ that *guarantee* a unique solution, so we focus on the condition $\Delta \neq 0$.

Q: Is there a faster way to calculate the determinant than expanding along the first row?

Yes, you can use row operations to create zeros in a row or column, which simplifies the expansion. For example, you could perform $R_2 \to R_2 - 2R_1$ and $R_3 \to R_3 - 3R_1$. This would make the first column $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, making the expansion much faster. However, for a $3 \times 3$ matrix, direct expansion is often just as quick and less prone to arithmetic errors if you are careful with signs.

Question

Let S be the set of all real values of k for which the system of linear equations $x + y + z = 2$, $2x + y - z = 3$, $3x + 2y + kz = 4$ has a unique solution. Then S is

Options:

(A) an empty set

(B) equal to R

(C) equal to {0}

(D) equal to R - {0}

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a 3D space, looking at three flat planes. Each equation in our system represents one of these planes.

When we ask for a unique solution, we are asking: at what point do these three planes intersect? If they intersect at exactly one point, the system is consistent and unique. If they are parallel or intersect in a line, the situation changes. This is the geometric soul of the problem.

### The Engine of the System

To solve this, we look at the coefficient matrix $A$. We extract the coefficients of $x$, $y$, and $z$ from each equation.

Our system is:
$x + y + z = 2$
$2x + y - z = 3$
$3x + 2y + kz = 4$

The matrix $A$ is defined as:
$$A = \begin{pmatrix} 1 & 1 & 1 \ 2 & 1 & -1 \ 3 & 2 & k \end{pmatrix}$$

This matrix is the engine of our system. If its determinant, $\Delta$, is non-zero, the system is guaranteed to have a unique solution. This is the power of Cramer's Rule.

### The Calculation

Now, let us calculate $\Delta$. We expand along the first row:
$$\Delta = 1(1 \cdot k - (-1) \cdot 2) - 1(2 \cdot k - (-1) \cdot 3) + 1(2 \cdot 2 - 1 \cdot 3)$$

Let us break this down carefully. The first term is $1(k + 2)$. The second term, remembering the negative sign, is $-1(2k + 3)$. The third term is $1(4 - 3) = 1$.

Combining these, we get:
$$\Delta = (k + 2) - (2k + 3) + 1$$

Simplifying this, we have:
$$\Delta = k + 2 - 2k - 3 + 1$$

Combining the like terms, $k - 2k = -k$ and $2 - 3 + 1 = 0$. Thus, we find:
$$\Delta = -k$$

### The Final Verdict

We know that for a unique solution, $\Delta 
eq 0$. Therefore, $-k 
eq 0$, which implies **$k 
eq 0$**.

This means that $k$ can be any real number except zero. The set $S$ of all such values is **$\mathbb{R} - \{0\}$**.

It is a beautiful result, isn't it? A single parameter $k$ determines the entire nature of the system. Keep this logic in your toolkit, and you will master any system of linear equations that comes your way.

Animated Solution for Mathematics - Matrices and Determinants: Let S be the set of all real values of k for which the system of linear equations $x + y + z = 2$ , $2 x + y - z = 3$ , $3 x + 2 y + k z = 4$ has a unique solution. Then S is

Select Answer:

Visualized Solution

Analyze the System of Equations

Condition for Unique Solution

Construct the Coefficient Matrix

Expand the Determinant $Δ$

Calculate the First Term

Calculate the Second Term

Calculate the Third Term

Simplify the Expression for $Δ$

Final Value of $Δ$

Apply the Unique Solution Condition

Conclusion and Final Set S

The Sigma Insight: Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

Analyzing the Setup

The Engine of the System

The Calculation

The Final Verdict

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