Let A=| 1 2 -1 |-2 -1 1

step 1 Hi student welcome to the solution we have a set of equations here from this the coefficient mat

Let $A=\left|\begin{array}{ccc}1 & 2 & -1 \\ \mid-2 & -1 & 1\end{array}\right|$ be the coefficient matrix of a homogeneous system in $\mathrm{x}, \mathrm{y}$, and $\mathrm{z}$. Solve this system to illustrate that a homogeneous system of 3 equations in the unknowns, $\mathrm{x}, \mathrm{y}, \mathrm{z}$ has a unique solution.

Key Concepts

Homogeneous System

A homogeneous system of linear equations is one in which all of the constant terms are zero. This property guarantees that the system always has at least one solution, called the trivial solution, where all the unknowns are set to zero. Homogeneous systems are central in linear algebra because they help illustrate concepts such as linear independence, the null space of a matrix, and conditions under which nontrivial solutions exist.

Coefficient Matrix

The coefficient matrix of a system of equations is a rectangular array containing only the coefficients of the variables in the system. It is used extensively in various methods such as Gaussian elimination or determining matrix rank and is pivotal in understanding the structure of the system, including properties like consistency, dependency, and the determination of solution uniqueness.

Trivial Solution

In the context of homogeneous systems, the trivial solution is the solution where every variable is equal to zero. This solution always exists regardless of the coefficients in the system. The trivial solution is important because its uniqueness or the existence of additional nontrivial solutions depends on the properties of the coefficient matrix, such as its rank.

Uniqueness of Solutions in Linear Systems

A homogeneous system of linear equations will have a unique solution (namely, the trivial solution) if the coefficient matrix has full rank, meaning that its rows or columns are linearly independent. This implies that the determinant of the coefficient matrix (if it is square) is nonzero. The concept underscores the fact that the structure of the matrix directly influences whether or not additional solutions exist beyond the trivial one.

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