Animated Solution for Mathematics - Matrices and Determinants: If the system of linear equations, $$\begin{gathered} x+y+z=6 \\ x+2y+3z=10 \\ 3x+2y+\lambda z=\mu \end{gathered}$$ has more than two solutions, then $\mu -{\lambda }^2$ is equal to . . . . .

In the world of linear algebra, three planes typically intersect at a single point. However, the problem specifies that the system has "more than two solutions."

In a linear system, this condition implies that the planes are not independent; they must intersect along a common line. Geometrically, this is analogous to the spine of an open book where all pages meet, resulting in an infinite number of solutions.

The Determinant

To ensure the system is dependent, the determinant $\Delta$ of the coefficient matrix must be zero. We define the determinant as follows:

Expanding this determinant along the first row, we obtain:

Simplifying the expression yields:

Thus, we find the first required value: $\lambda =1$.

With $\lambda =1$, the third plane passes through the intersection line of the first two. To determine $\mu$, we identify a point that lies on this line by setting $z=0$.

The first two equations simplify to:

Subtracting the first equation from the second gives $y=4$. Substituting $y=4$ back into the first equation, we find $x=2$. Our point $P$ is $(2,4,0)$.

Since point $P(2,4,0)$ lies on the intersection line, it must satisfy the third equation: $3x+2y+\lambda z=\mu$. Substituting our known values $x=2,y=4,z=0,$ and $\lambda =1$:

The problem asks for the value of $\mu -{\lambda }^2$. Substituting our calculated values:

The final result is $13$.