Animated Solution for Mathematics - Matrices and Determinants: The number of real values $\lambda$, such that the system of linear equations $2x-3y+5z=9$, $x+3y-z=-18$, $3x-y+({\lambda }^2-|\lambda |)z=16$ has no solution, is :-

A Journey into Linear Systems

Welcome, future engineer! Today, we are not just solving a system of linear equations; we are exploring the architecture of space itself. When you look at a system of three linear equations, you are looking at three planes in three-dimensional space.

Usually, these planes intersect at a single, beautiful point—a unique solution. But what happens when they refuse to meet? What happens when the system is 'inconsistent'? That is the mystery we are unraveling today.

The Geometry of the Problem

We are given the system:

The question asks for the number of real values of $\lambda$ that lead to 'no solution.' In the language of linear algebra, this is a call to Cramer's Rule.

We know that for a system to have a unique solution, the determinant of the coefficient matrix, $D$, must be non-zero. Conversely, if $D=0$, the system is either inconsistent (no solution) or dependent (infinite solutions).

To ensure there is absolutely no solution, we must satisfy two conditions: first, $D=0$, and second, at least one of the auxiliary determinants ($D_x,D_y,\text{ or }{D}_z$) must be non-zero. This is our roadmap.

The Determinant Dance

Let us construct our main determinant $D$. We extract the coefficients of $x,y,\text{ and }z$ from the equations:

Now, we expand this along the first row. Take a deep breath—precision is key here. We calculate:

Simplifying this step-by-step is where the magic happens. Distributing the terms, we get:

Combining the terms, we arrive at a beautifully compact expression:

The Quadratic Trap

We set $D=0$ to find the critical values of $\lambda$:

This looks intimidating, but let us use a substitution to tame it. Let $t=|\lambda |$. Since ${\lambda }^2=|\lambda {|}^2$, we can rewrite this as a quadratic equation in $t$:

Multiplying by 9 to clear the fraction, we get $9t^2-9t-43=0$. Applying the quadratic formula, $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, we find:

Here is the moment of truth. We have two potential values for $t$. However, recall that $t=|\lambda |$, and the absolute value of a real number must be non-negative.

The root $\frac{9-\sqrt{1629}}{18}$ is clearly negative because $\sqrt{1629}>9$. We must reject it! This leaves us with exactly one valid positive value for $t$:

The Final Verification

Since $|\lambda |=k$ (where $k$ is a positive constant), we have two possible values for $\lambda$: $\lambda =k$ and $\lambda =-k$. This gives us two potential candidates.

But are we done? Not quite! We must ensure that for these values, the system is truly inconsistent. We check $D_z$:

Calculating this determinant yields $180$, which is definitely not zero. Since $D=0$ and $D_z eq 0$, the system is indeed inconsistent for these values of $\lambda$.

We have navigated the algebra, respected the geometric constraints, and verified our conditions. We found that there are exactly two real values of $\lambda$ that satisfy the condition for no solution. You have successfully mastered the logic of linear systems.