Animated Solution for Mathematics - Matrices and Determinants: If S is the set of distinct values of 'b' for which the following system of linear equations $x+y+z=1$, $x+ay+z=1$, $ax+by+z=0$ has no solution, then S is:

My dear student, welcome to the fascinating world of linear algebra. Today, we are not just solving a system of equations; we are exploring the geometric soul of three-dimensional space.

Imagine you are standing in a room where three large, flat sheets of glass represent our equations: $x+y+z=1$ $x+ay+z=1$ $ax+by+z=0$

Our mission is to find the value of $b$ that makes these planes refuse to meet. This is the condition of 'no solution'.

The Determinant

To begin, we must ask: when does a system of three equations fail to have a unique solution? The answer lies in the determinant of the coefficient matrix, which we denote as $\Delta$.

If $\Delta eq 0$, the planes intersect at a single, beautiful point. But we want chaos, we want no intersection! So, we must force $\Delta =0$.

Let us write down our matrix:

Expanding this along the first row, we get:

Simplifying this, we see the terms $a-b-1+a+b-a^2$. Notice how the $b$ and $-b$ terms vanish into thin air! We are left with $-a^2+2a-1$, which is simply $-(a-1{)}^2$.

Setting this to zero, we find that $a$ must be $1$. This is our first major breakthrough.

When Planes Align

Now, let us substitute $a=1$ back into our original system. The equations become: $x+y+z=1$ $x+y+z=1$ $x+by+z=0$

Look closely! The first two equations are identical. They represent the exact same plane.

Our system has collapsed from three planes down to just two distinct planes: $x+y+z=1$ and $x+by+z=0$. For there to be no solution, these two remaining planes must be strictly parallel. They must be like two train tracks that run forever without ever touching.

Parallelism

How do we ensure two planes are parallel? We look at their normal vectors. For planes $A_{1}x+B_{1}y+C_{1}z=D_1$ and $A_{2}x+B_{2}y+C_{2}z=D_2$, they are parallel if:

Applying this to our system, we have:

The equality $\frac{1}{1}=\frac{1}{b}$ immediately tells us that $b=1$. Checking the constant terms, $1 eq 0$ is a true statement, confirming that the planes are indeed distinct and parallel.

We have arrived at our destination. The set $S$ of values for $b$ is simply the singleton set $\{1\}$. It is elegant, it is precise, and it is the only way to satisfy the condition of no solution.