Determine all values of the constants a and b for which the...

Determine all values of the constants $a$ and $b$ for which the following system has (a) no solution, (b) an infinite number of solutions, and (c) a unique solution.
$$
\begin{aligned}
x_{1}-\quad & a x_{2}=3 \\
2 x_{1}+\quad & x_{2}=6 \\
-3 x_{1}+(a+b) x_{2} &=1
\end{aligned}
$$

Key Concepts

Consistency and Rank

The concept of consistency in a system of linear equations is determined by comparing the rank of the coefficient matrix with the rank of the augmented matrix. A system is consistent—meaning it has at least one solution—if and only if the rank of the coefficient matrix equals the rank of the augmented matrix. If the augmented matrix has a higher rank than the coefficient matrix, the system is inconsistent (no solution).

Parameter Conditions in Linear Systems

In systems where coefficients depend on parameters, the outcome of the system (unique solution, infinite solutions, or no solution) can change based on the values of these parameters. Analyzing how parameters affect the determinant and the rank of the matrices is essential for determining the conditions under which the system behaves in each way.

Systems of Linear Equations

A system of linear equations is a collection of one or more equations involving the same set of variables. The solution to such a system is a set of variable values that simultaneously satisfy all the equations. The system can have a unique solution, infinitely many solutions, or no solution, depending on the relationships between the equations.

Determinant and Invertibility

The determinant of a square coefficient matrix is a crucial tool in understanding the nature of a linear system. A nonzero determinant indicates that the matrix is invertible, which in turn guarantees a unique solution for the system. If the determinant is zero, the matrix is singular, and the system may either have infinitely many solutions or be inconsistent.

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