Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $2x+y-z=5$, $2x-5y+\lambda z=\mu$, $x+2y-5z=7$ has infinitely many solutions, then $(\lambda +\mu {)}^2+(\lambda -\mu {)}^2$ is equal to

A Journey into Linear Systems

Welcome, future engineer! Today, we are going to peel back the curtain on one of the most elegant concepts in linear algebra: the system of equations.

When you see a system like $2x+y-z=5$, $2x-5y+\lambda z=\mu$, and $x+2y-5z=7$, don't just see a collection of numbers. See a story. See three planes dancing in three-dimensional space.

Visualizing the Planes

Imagine you are standing in a vast, empty room. You have three giant, flat sheets of glass—these are your planes.

In a typical scenario, these three planes might intersect at a single point, like the corner of a room. But the problem tells us something special: this system has infinitely many solutions.

Geometrically, this means the planes are not just meeting at a point; they are all passing through a single, common line. Think of the pages of an open book meeting at the spine. That spine is our line of infinite solutions.

The Algebra of Destiny

How do we force this harmony? We turn to the master key: Cramer's Rule.

For a system to have infinitely many solutions, the main determinant of the coefficient matrix, $\Delta$, must be zero. But that is only half the battle.

If $\Delta =0$, the system could be inconsistent—meaning the planes never meet at all. To ensure they meet at a line, we must also ensure that the numerator determinants, ${\Delta }_1,{\Delta }_2,$ and ${\Delta }_3$, are all zero. This is the condition of dependency.

Let's construct our main determinant $\Delta$ using the coefficients of $x,y,$ and $z$:

Expanding this along the first row, we get:

This simplifies to:

Expanding the terms, we find:

Combining like terms, we arrive at $51-3\lambda =0$, which gives us the elegant result $\lambda =17$. We have found our first missing piece!

The Final Calculation

Now that we have $\lambda$, we need $\mu$. We turn to ${\Delta }_3$, where we replace the $z$-column with our constants: $5,\mu ,$ and $7$.

Expanding this determinant along the first row:

Simplifying this, we get $-39-3\mu =0$, leading us directly to $\mu =-13$.

With $\lambda =17$ and $\mu =-13$, we are ready for the final act. The question asks for the value of $(\lambda +\mu {)}^2+(\lambda -\mu {)}^2$.

Substituting our values:

And there it is! The geometry and the algebra have converged perfectly. Remember, whenever you face these systems, visualize the planes, trust the determinants, and keep your calculations precise. You have mastered the system!