Animated Solution for Mathematics - Matrices and Determinants: The system of linear equations $x+y+z=6$, $2x+5y+az=36$, $x+2y+3z=b$ has

The system of linear equations is given by: 1) $x+y+z=6$ 2) $2x+5y+az=36$ 3) $x+2y+3z=b$

To determine the nature of the solutions, we examine the determinant of the coefficient matrix, $\Delta$. This determinant acts as the "DNA" of the system, dictating whether a unique solution exists or if the system is singular.

We define the coefficient matrix and calculate its determinant:

Expanding along the first row:

Simplifying the expression:

For the system to have infinitely many solutions, the determinant must be zero. Setting $\Delta =0$ yields $8-a=0$, which gives us the first key: $a=8$.

With $a=8$, we substitute this value back into the system: 1) $x+y+z=6$ 2) $2x+5y+8z=36$ 3) $x+2y+3z=b$

We perform row operations to analyze the consistency of the system. First, we eliminate $x$ from the second equation using $R_2\to R_2-2R_1$:

Next, we eliminate $x$ from the third equation using $R_3\to R_3-R_1$:

We are left with two derived equations: $y+2z=8$ and $y+2z=b-6$. For the system to possess infinitely many solutions, these two equations must be consistent and represent the same line.

This requires the constants on the right-hand side to be equal:

Thus, the values that unlock the system are $a=8$ and $b=14$. At these values, the equations become linearly dependent, resulting in infinitely many solutions.