Animated Solution for Mathematics - Matrices and Determinants: The system of linear equations $x+y+z=2$, $2x+3y+2z=5$, $2x+3y+(a^2-1)z=a+1$

My dear students, welcome to the heart of linear algebra. Today, we are not just solving a system of equations; we are investigating the very nature of existence—or in this case, non-existence.

We are given a system of three linear equations: $x+y+z=2$ $2x+3y+2z=5$ $2x+3y+(a^2-1)z=a+1$

Our mission is to uncover the values of $a$ that render this system inconsistent. In the language of geometry, this means we are looking for the scenario where these three planes refuse to meet at a single point, or even a line.

The Gatekeeper—The Determinant $D$

To understand the behavior of any system of linear equations, we must first consult the 'Gatekeeper': the determinant of the coefficient matrix, $D$. This value tells us whether the system has a unique solution.

If $D eq 0$, the system is well-behaved, and a unique solution exists. But we are hunting for inconsistency, which forces us to look where $D=0$. Let us construct our matrix:

Expanding this along the first row is our most reliable path. We calculate:

Notice the elegance here—the third term vanishes! Simplifying this, we get:

There it is. The condition for our system to lose its unique solution is $a^2-3=0$, or $|a|=\sqrt{3}$. This is our critical bifurcation point.

The Truth Teller—The Auxiliary Determinant $D_3$

Now, here is where many students stumble. They assume that $D=0$ automatically means the system is inconsistent. But wait! $D=0$ is a necessary condition for inconsistency, but it is not sufficient.

It could also mean the system has infinitely many solutions. To distinguish between these two, we must check the auxiliary determinant, $D_3$ (or $D_z$). We replace the third column of our matrix with the constants from the right-hand side of our equations:

We could expand this directly, but let us be smarter. Let us use the power of row operations. If we perform $R_3\to R_3-R_2$, we get:

This is the beauty of linear algebra! By creating those two zeros in the third row, the expansion becomes trivial. Expanding along the third row, we get:

The Final Synthesis

We have our two critical pieces of information: $D=a^2-3$ and $D_3=a-4$. For the system to be inconsistent, we require $D=0$ AND $D_3 eq 0$.

When $|a|=\sqrt{3}$, we know $D=0$. Now, let us check $D_3$ at these points:

If $a=\sqrt{3}$, then $D_3=\sqrt{3}-4$, which is clearly not zero.

If $a=-\sqrt{3}$, then $D_3=-\sqrt{3}-4$, which is also not zero.

Since $D=0$ and $D_3 eq 0$, the system is undeniably inconsistent. We have navigated the logic, avoided the traps, and arrived at the truth. The system is inconsistent for $a=\pm \sqrt{3}$.