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

Imagine you are standing in a vast, three-dimensional room. You have three large, flat sheets of paper—these are your planes. Each equation in our system, $x+y+z=6$, $x+2y+3z=10$, and $3x+2y+\lambda z=\mu$, defines the orientation of one of these sheets.

Usually, three planes intersect at a single point, like the corner of a room. However, the problem states that the system has "more than two solutions." Geometrically, this is a signal.

Three planes cannot intersect at exactly three points. They either meet at one point, zero points, or they share an entire line. If they share a line, they share an infinite number of points. Thus, our system is a geometric dance where all three planes must align along a common axis.

To solve this, we turn to the elegant machinery of linear algebra: Cramer's Rule. For a system to have infinite solutions, the main determinant of the coefficient matrix, $D$, must vanish.

If $D eq 0$, the system would have a unique solution. But we need more; we need the planes to be consistent. This requires the auxiliary determinants—$D_x$, $D_y$, and $D_z$—to also be zero.

Think of $D=0$ as the condition for the planes to be "parallel" in a sense, and $D_z=0$ as the condition that ensures they do not just drift apart into a prism, but actually lock together.

Let us construct our main determinant $D$ using the coefficients of $x,y,$ and $z$:

Expanding along the first row, we get:

Simplifying this, we find $2\lambda -6-\lambda +9-4=0$, which reduces beautifully to $\lambda -1=0$. This gives us our first victory: $\lambda =1$.

Now that we know $\lambda =1$, we must find $\mu$. We use the condition $D_z=0$. We replace the $z$-column of our coefficient matrix with the constants from the right-hand side of our equations: $6,10,$ and $\mu$.

Expanding this, we get:

This simplifies to $2\mu -20-\mu +30-24=0$, which leaves us with $\mu -14=0$, or $\mu =14$.

With $\lambda =1$ and $\mu =14$ in hand, the final step is a simple calculation:

We have navigated the geometry, applied the algebra, and arrived at the truth. The system is consistent, the planes are locked, and the final answer is 13.