Animated Solution for Mathematics - Matrices and Determinants: The values of $a$ and $b$, for which the system of equations $2x+3y+6z=8$, $x+2y+az=5$, $3x+5y+9z=b$ has no solution, are:

Imagine you are standing in a three-dimensional space with three planes defined by the following linear equations:

In linear algebra, these equations represent physical surfaces. While these planes typically intersect at a single point, we are investigating the condition for no solution, where the planes never meet at a common point.

To unlock this mystery, we utilize the machinery of Cramer's Rule. We define the main determinant of the coefficient matrix, $\Delta$, as:

For a system to have no solution, the first requirement is that the system must be singular, meaning $\Delta =0$. If $\Delta eq 0$, the system would possess a unique solution.

However, $\Delta =0$ is a necessary but not sufficient condition. We must also ensure the system is not 'dependent' (which would lead to infinite solutions) by verifying that at least one of the numerator determinants—${\Delta }_x$, ${\Delta }_y$, or ${\Delta }_z$—is non-zero.

Unveiling 'a'

Let us expand $\Delta$ along the first row:

Performing the algebraic expansion:

Combining the terms, we find:

Setting this to zero, we immediately see that $a=3$. This is the value that renders our system singular.

The Condition for 'b'

To ensure the system is truly inconsistent, we construct ${\Delta }_z$ by replacing the third column of the coefficient matrix with the constants from the right side of our equations:

Expanding this along the first row:

Simplifying this expression:

This collapses into:

For the system to have no solution, we must satisfy the condition $\Delta_z eq 0$. Therefore, $b eq 13$.

By setting $\Delta =0$, we forced the system to be singular, yielding $a=3$. By ensuring $\Delta_z eq 0$, we guaranteed that the system is inconsistent rather than dependent, yielding $b eq 13$.

The values for which the system has no solution are $a=3$ and $b eq 13$. This result describes a configuration where the planes exist in space while forever avoiding a common intersection.