Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z =

Q: Wait, why do we reject the negative root here? Or rather, why do we care about $D \neq 0$?

Great question! In the world of linear algebra, the determinant $D$ acts as a gatekeeper. If $D \neq 0$, the system is 'well-behaved' and has a unique intersection point. If $D = 0$, the system collapses—it either has no solution (the planes never meet) or infinitely many solutions (the planes meet at a line or are coincident). Since the problem explicitly demands a 'unique solution,' we must strictly avoid the $D = 0$ scenario.

Q: Can I use Gaussian elimination instead of determinants? Is it faster?

You absolutely could use Gaussian elimination, but be careful! While it is a powerful tool, it often leads to messy algebraic expressions when variables like $n$ are involved. Determinants are usually more elegant for checking the *existence* of a unique solution because they condense the entire system's behavior into a single scalar value. Stick to the determinant method here—it is the path of least resistance.

Q: What if $D=0$? Does that mean no solution?

Not necessarily! If $D=0$, the system is either inconsistent (no solution) or dependent (infinitely many solutions). However, for this specific problem, we don't need to distinguish between those two cases. We only care that the solution is *not* unique. Therefore, $D=0$ is our 'forbidden zone' regardless of whether it leads to zero or infinite solutions.

Question

Let $n$ be the number obtained on rolling a fair die. If the probability that the system $x - ny + z = 6$, $x + (n - 2)y + (n + 1)z = 8$, $(n - 1)y + z = 1$ has a unique solution is $\frac{k}{6}$, then the sum of $k$ and all possible values of $n$ is :

Options:

(A) 20

(B) 24

(C) 21

(D) 22

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are not just solving a system of equations; we are exploring the delicate balance of 3D space. Imagine three planes floating in space.

Usually, they intersect at a single point—a unique solution. But sometimes, they align in ways that make that point vanish or expand into a line. Our goal is to find when that 'uniqueness' holds true, governed by the roll of a fair die.

### The Determinant as a Gatekeeper

We are given a system of three linear equations:

1. $x - ny + z = 6$
2. $x + (n - 2)y + (n + 1)z = 8$
3. $(n - 1)y + z = 1$

When we look at this, we shouldn't just see variables. We should see a coefficient matrix $A$. The condition for a unique solution is the heartbeat of linear algebra: the determinant of the coefficient matrix must be non-zero.

If $|A| = 0$, the system is singular, and uniqueness is lost. So, our mission is to find $|A|$ and ensure it is not zero.

### Constructing the Matrix

Let us extract the coefficients carefully. For the first equation, the coefficients are $1, -n, 1$. For the second, $1, (n-2), (n+1)$.

For the third, notice the absence of $x$, so the coefficient is $0$, followed by $(n-1)$ and $1$. Our determinant $D$ looks like this:

$$D = \begin{vmatrix} 1 & -n & 1 \ 1 & n-2 & n+1 \ 0 & n-1 & 1 \end{vmatrix}$$

### The Algebraic Dance

Expanding this determinant is where the magic happens. We choose the first column because of that beautiful zero in the third row. It simplifies our expansion significantly:

$$D = 1 \cdot \begin{vmatrix} n-2 & n+1 \ n-1 & 1 \end{vmatrix} - 1 \cdot \begin{vmatrix} -n & 1 \ n-1 & 1 \end{vmatrix} + 0$$

Let's break this down. The first minor is $(n-2)(1) - (n-1)(n+1)$. Remember your algebraic identities! $(n-1)(n+1)$ is a difference of squares: $n^2 - 1$.

So, the first part becomes:

$$(n - 2) - (n^2 - 1) = n - 2 - n^2 + 1 = -n^2 + n - 1$$

Now, the second minor. We have $-1$ multiplied by the determinant of the second minor:

$$-1 \cdot [(-n)(1) - (n-1)(1)] = -1 \cdot [-n - n + 1] = -1 \cdot [-2n + 1] = 2n - 1$$

Combining these, we get:

$$D = (-n^2 + n - 1) + (2n - 1) = -n^2 + 3n - 2$$

### The Filter of Constraints

We need $D 
eq 0$. Let's factorize our expression:

$$D = -(n^2 - 3n + 2) = -(n - 1)(n - 2)$$

For $D$ to be non-zero, $n$ cannot be $1$ and $n$ cannot be $2$. Since $n$ is the outcome of a fair die, $n \in \{1, 2, 3, 4, 5, 6\}$.

By removing the forbidden values, we are left with the valid set: $n \in \{3, 4, 5, 6\}$.

### Final Calculation

There are $4$ favorable outcomes out of $6$ possible rolls. Thus, the probability is $\frac{4}{6}$. The problem states this is $\frac{k}{6}$, so $k = 4$.

Finally, we calculate the sum of $k$ and all valid values of $n$:

$$Sum = k + (3 + 4 + 5 + 6) = 4 + 18 = \mathbf{22}$$

And there you have it! Through the lens of determinants, we navigated the constraints and found our answer. Keep this logical rigor in your toolkit—it will serve you well in every challenge that follows.

Select Answer:

Visualized Solution

System of Equations & Die Roll

Condition for Unique Solution

Forming the Determinant $D$

Expanding Along Column $1$

Evaluating the First Minor

Evaluating the Second Minor

Combining to Find $D$

Factorizing the Determinant

Valid Outcomes for Unique Solution

Calculating Probability and $k$

Final Sum Calculation

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

Analyzing the Setup

The Determinant as a Gatekeeper

Constructing the Matrix

The Algebraic Dance

The Filter of Constraints

Final Calculation

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