Animated Solution for Mathematics - Matrices and Determinants: Let the system of equations $x+2y+3z=5,2x+3y+z=9,4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda +2\mu$ is equal to :

Imagine you are standing in a vast, three-dimensional space. You have three planes, each defined by a linear equation: $x+2y+3z=5$, $2x+3y+z=9$, and $4x+3y+\lambda z=\mu$.

In a typical scenario, these three planes would slice through each other and meet at a single, unique point $(x,y,z)$. However, this system has an infinite number of solutions.

This implies that the planes are not meeting at a single point; they are intersecting along a common line. Every point on that line is a valid solution.

To unlock this mystery, we turn to Cramer's Rule. For a system to have infinite solutions, the main determinant of the coefficient matrix, denoted by $\Delta$, must be zero.

This indicates that the equations are linearly dependent. To ensure the system is consistent and not simply composed of parallel planes that never meet, we must also ensure that the auxiliary determinants—${\Delta }_1$, ${\Delta }_2$, and ${\Delta }_3$—are also zero.

This condition guarantees that our planes are locked together in an infinite line of intersection.

Unmasking $\lambda$

Let us construct our main determinant $\Delta$ using the coefficients of our variables:

We expand this along the first row:

Simplifying the expression, we get $3\lambda -3-4\lambda +8-18$. Combining the terms, we arrive at $\Delta =-\lambda -13$.

Since we require infinite solutions, we set $\Delta =0$, which gives us $-\lambda -13=0$, or simply $\lambda =-13$.

Finding $\mu$

Now that we know $\lambda =-13$, we turn our attention to ${\Delta }_1$. We replace the first column of our determinant with the constants from the right side of our equations: $5,9,\mu$.

Expanding this along the first row:

This simplifies to:

Grouping the terms, we get $105-7\mu$. Setting ${\Delta }_1=0$ leads us to $7\mu =105$, which means $\mu =15$.

We have navigated the geometry and the algebra to find $\lambda =-13$ and $\mu =15$. The problem asks for the value of $\lambda +2\mu$.

Substituting our values:

The final result is $17$. Whenever you encounter 'infinite solutions' in a $3\times 3$ system, remember the condition $\Delta ={\Delta }_1={\Delta }_2={\Delta }_3=0$.