Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $$\begin{gathered} x+y+z=2 \\ 2x+4y-z=6 \\ 3x+2y+\lambda z=\mu \end{gathered}$$ has infinitely many solutions, then:

Imagine you are standing in a 3D coordinate system with three planes. Usually, these planes intersect at a single point, providing a unique solution for $x$, $y$, and $z$.

However, we are looking for a scenario where these planes intersect along a common line, creating an infinite number of solutions. To find the values of $\lambda$ and $\mu$ that allow this, we utilize the tool of Cramer's Rule.

Our first step is to analyze the core structure of the system. We construct the main coefficient determinant, $\Delta$, using the coefficients of $x$, $y$, and $z$:

For the system to have infinitely many solutions, this determinant must vanish, meaning $\Delta =0$. Expanding this along the first row:

Simplifying this, we get $\Delta =(4\lambda +2)-(2\lambda +3)-8$, which collapses into $\Delta =2\lambda -9$. Setting this to zero, we find our first key:

Now that we have $\lambda$, we need to find $\mu$. We invoke the condition that the auxiliary determinant ${\Delta }_x$ must also be zero.

We construct ${\Delta }_x$ by replacing the first column of our original matrix with the constant terms from the right side of our equations: $2,6,\mu$. Substituting $\lambda =\frac{9}{2}$, we get:

Expanding this along the first row, we calculate:

This simplifies to ${\Delta }_x=2(18+2)-(27+\mu )+(12-4\mu )$. Further reduction yields:

Setting ${\Delta }_x=0$, we find $5\mu =25$, which results in $\mu =5$.

We have determined the values: $\lambda =\frac{9}{2}$ and $\mu =5$.

Testing the condition $2\lambda +\mu =14$, we substitute our values:

The result matches perfectly. We have successfully navigated the complexity of linear systems to identify the constraints that define the geometry of the intersection.