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

Imagine you are standing in a three-dimensional space, looking at three planes defined by the equations:

These planes are like sheets of glass floating in space, and their intersection is what we call the solution to our system. The parameter $\lambda$ shifts these planes, and our mission is to find the specific values of $\lambda$ that cause these planes to either never meet or to overlap in a way that creates a line or plane of intersection.

To unlock this mystery, we turn to the most powerful tool in our linear algebra arsenal: the determinant of the coefficient matrix, $D$. Think of $D$ as the 'volume' or 'scaling factor' of the transformation defined by our system.

If $D eq 0$, the system is well-behaved and has a unique solution. When $D=0$, the system collapses, and we enter the realm of inconsistency or dependency. Let us calculate $D$:

Expanding this along the first row, we get:

Simplifying this expression, we find:

Now, we set $D=0$ to find the values of $\lambda$ where the system's behavior changes. The equation $-{\lambda }^2-6\lambda +16=0$ is equivalent to:

Factoring this quadratic, we look for two numbers that multiply to $-16$ and add to $6$. Those numbers are $8$ and $-2$. Thus:

This gives us the critical values $\lambda =2$ and $\lambda =-8$.

Let us test the case where $\lambda =2$. We know $D=0$, but we must determine if the system is inconsistent or dependent. We calculate $D_1$ by replacing the first column of the coefficient matrix with the constants $5,8,10$:

Expanding this, we get:

Since $D_1 = 16 eq 0$, the system is inconsistent. This means that when $\lambda =2$, the planes do not share a common point of intersection.

The algebraic conclusion is clear: there is no solution for this value. By systematically checking the determinant, you have proven that for $\lambda =2$, the system is inconsistent.