Animated Solution for Mathematics - Matrices and Determinants: The system of linear equations $$\begin{gathered} \lambda x+2y+2z=5 \\ 2x+3y+5z=8 \\ 4x+\lambda y+6z=10 \end{gathered}$$ has

A Journey into $\lambda$

Welcome, future engineers! Today, we are not just solving a system of linear equations; we are exploring the delicate balance of three-dimensional space.

When you look at a system like this, I want you to stop seeing just rows of numbers. I want you to see three planes in space. The question is: how do these planes interact? Do they meet at a single point, do they form a line of intersection, or do they refuse to meet at all?

The Gatekeeper - The Determinant $\Delta$

Our journey begins with the coefficient matrix. We define the determinant $\Delta$ as:

Why do we care about $\Delta$? Think of $\Delta$ as the 'volume' of the parallelepiped formed by the normal vectors of these planes.

If $\Delta eq 0$, the volume is non-zero, meaning the planes are linearly independent and intersect at a unique point. But if $\Delta =0$, the volume collapses. The planes are now linearly dependent, and this is the moment where the system becomes 'singular.'

The Algebraic Grind

Now, let's expand this determinant. Precision is our best friend here. Expanding along the first row:

Let's break this down. The first term is $\lambda (18-5\lambda )$. The second term is $-2(12-20)$, which simplifies to $-2(-8)=16$. The third term is $2(2\lambda -12)$.

Combining these, we get:

Factoring this expression, we obtain:

This gives us our critical values: $\lambda =4$ and $\lambda =0.4$. These are the values where the 'volume' of our system collapses. At these points, the system is no longer unique.

The Verdict - Cramer's Rule in Action

Now, let us test $\lambda =2$ (as per the provided example). We need to know: does the system have no solution or infinite solutions? We turn to Cramer's Rule.

We calculate ${\Delta }_x$ by replacing the first column of the coefficient matrix with the constants from the right-hand side of our equations: $5,8,10$.

Let's expand this. It is just arithmetic now, but keep your focus sharp:

Look at what we have found! For $\lambda =2$, we have $\Delta =0$, but ${\Delta }_x=16$.

Because $\Delta_{x} eq 0$, the system is inconsistent. Geometrically, this means the planes do not share any common point of intersection.

They might be parallel, or they might form a triangular prism structure, but they never meet. Thus, we conclude with confidence: the system has no solutions when $\lambda =2$.