Consider the following system of linear equations: x+2 y+z=1 2 x+y+z=α4 x+5 y+3 z=α^2. Then the

step 1 In the problem, given system can be written as 121211453 xY z that is equal to 1 alpha alpha squ

Consider the following system of linear equations: x+2 y+z=1...

Consider the following system of linear equations: $x+2 y+z=1$ $2 x+y+z=\alpha$ $4 \mathrm{x}+5 \mathrm{y}+3 \mathrm{z}=\alpha^{2}$. Then the system has (a) infinitely many solutions when $\alpha=-1$ or 2 (b) infinitely many solutions when $\alpha=1$ or $-2$ (c) no solution when $\alpha \in \mathrm{R}-\{-1,2\}$ (d) no solution when $\alpha \in \mathrm{R}-\{1,-2\}$

Consider the following system of linear equations:
$x+2 y+z=1$
$2 x+y+z=\alpha$
$4 \mathrm{x}+5 \mathrm{y}+3 \mathrm{z}=\alpha^{2}$. Then the system has
(a) infinitely many solutions when $\alpha=-1$ or 2
(b) infinitely many solutions when $\alpha=1$ or $-2$
(c) no solution when $\alpha \in \mathrm{R}-\{-1,2\}$
(d) no solution when $\alpha \in \mathrm{R}-\{1,-2\}$

Key Concepts

Rank of a Matrix and the Rouché–Capelli Theorem

The rank of a matrix is a measure of the number of linearly independent rows or columns it contains. The Rouché–Capelli theorem states that a linear system has a solution if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix. This theorem provides a systematic way to determine whether a system is consistent, and if consistent, whether the solution is unique or infinite.

System of Linear Equations

This concept refers to a collection of one or more linear equations involving the same set of variables. In linear algebra, such systems are studied to determine the values of the variables that satisfy all the equations simultaneously, and they are the foundation for methods like matrix representation and Gaussian elimination.

Consistency and Inconsistency of Systems

A system is considered consistent if there exists at least one solution and inconsistent if no solution exists. The analysis of a system’s consistency is essential in understanding the conditions under which the equations can be satisfied, often utilizing the concepts of row reduction and the comparison of the rank of the coefficient matrix with that of the augmented matrix.

Parametric Systems

In many instances, systems of linear equations include parameters in their coefficients or constant terms. The inclusion of parameters means that the nature of the solutions—whether there are unique solutions, infinitely many solutions, or no solution—can depend on the specific values of these parameters. Analyzing parametric systems helps in understanding how changes in the parameters influence the behavior of the system.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by transforming the system into a simpler form, typically an upper triangular or row echelon form. This process facilitates the identification of the basic and free variables in the system, and it is a practical tool for assessing the consistency and number of solutions in linear systems.

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