Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $2x+3y-z=5$, $x+\alpha y+3z=-4$, $3x-y+\beta z=7$ has infinitely many solutions, then $13\alpha \beta$ is equal to

Imagine standing in a room where three walls meet. Each wall is a plane, and their intersection is a single point.

But what if those walls were arranged differently? What if they all met at a single, shared edge, like the pages of an open book meeting at the spine?

This is the geometric reality of a system of equations with infinitely many solutions. We are not looking for a single point $(x,y,z)$; we are looking for the entire line where these three planes coincide.

When three planes intersect in a common line, they are not independent. One plane is essentially a 'shadow' or a combination of the other two.

Mathematically, we can express one plane's equation as a linear combination of the others: $L_1=k_1{L}_2+k_2{L}_3$. This is a beautiful shortcut that bypasses the tedious determinant method.

We define our planes as:

By setting $L_1=k_1{L}_2+k_2{L}_3$, we are essentially saying that the first plane is a weighted sum of the other two.

Now, we substitute the expressions and compare the coefficients of $x,y,z$ and the constant terms.

For $x$, we have $2=k_1(1)+k_2(3)$, which gives us our first equation:

For the constant terms, we have $-5=k_1(4)+k_2(-7)$, or:

Solving this simple system is our next step. From the first, $k_1=2-3k_2$. Substituting this into the second, we get:

Thus, $k_2=\frac{13}{19}$ and $k_1=-\frac{1}{19}$.

With $k_1$ and $k_2$ in hand, finding $\alpha$ and $\beta$ becomes a game of matching coefficients.

For $y$, we have $3=k_1\alpha -k_2$. Substituting our values:

Multiplying by $19$ gives $57=-\alpha -13$, so $\alpha =-70$.

For $z$, we have $-1=3k_1+\beta k_2$. Substituting:

Multiplying by $19$ gives $-19=-3+13\beta$, so $13\beta =-16$.

We are asked for $13\alpha \beta$. We can write this as $\alpha \times (13\beta )$.

Substituting our values:

The beauty of this problem lies in how the geometry dictates the algebra. By visualizing the planes, we turned a complex system into a straightforward comparison of coefficients.

The final answer is 1120.